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# Quantum equilibration, thermalization and prethermalization in ultracold atoms

## Abstract

Over the past decade, there has been remarkable progress in our understanding of equilibration, thermalization and prethermalization, due in large part to experimental breakthroughs in ultracold atomic gases. These advances have made it possible to investigate how isolated quantum systems thermalize and why certain special many-body states do not. An overview on recent theoretical and experimental developments is given.

## Key points

• Equilibration, thermalization and prethermalization are universal phenomena that occur through loss of memory about the initial state of a system. Quantum thermalization proceeds through formation of entanglement: a subsystem can appear thermal if it is sufficiently entangled with the rest of the system.

• Ultracold atomic gases offer an ideal testbed for the fundamental study of relaxation, thermalization and prethermalization of isolated quantum systems because they can be almost perfectly isolated from surrounding environments or subjected to controlled dissipation.

• A number of remarkable experiments on fundamental aspects of thermalization have been reported in ultracold atomic systems, for which all microscopic details are known and can be controlled to high precision. This is a field where theory and experiment cross-fertilize.

• A many-body scar may be viewed as a many-body dark state, and belongs to a broad class of those states that are metastable in dissipative non-equilibrium situations. Such states are not thermal, but may be prethermal.

• Arguably, the most challenging problem is to identify a class of robust non-equilibrium states in an open dissipative environment. Many long-lived many-body systems could belong to this class.

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## References

1. Langen, T., Geiger, R. & Schmiedmayer, J. Ultracold atoms out of equilibrium. Annu. Rev. Condens. Matter Phys. 6, 201–217 (2015).

2. Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).

3. Mori, T., Ikeda, T. N., Kaminishi, E. & Ueda, M. Thermalization and prethermalization in isolated quantum systems: a theoretical overview. J. Phys. B 51, 112001 (2018).

4. Boltzmann, L. Über die Beziehung swischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über des Wärmegleichgewicht. Wien. Ber. 76, 373–435 (1877).

5. Tasaki, H. Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems. J. Stat. Phys. 163, 937–997 (2016).

6. Maxwell, J. C. Theory of Heat (Longman, 1871).

7. Gibbs, J. W. Elementary Principles in Statistical Mechanics (C. Scribner’s Sons, 1902).

8. Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 17, 549–560 (1905).

9. Einstein, A. Zum gegenwärtigen Stand des Strahlungsproblems. Phys. Z. 10, 185–193 (1909).

10. Kubo, R. Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn 12, 570–586 (1957).

11. Evans, D. J., Cohen, E. G. D. & Morriss, G. P. Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404 (1993).

12. Gallavotti, G. & Cohen, E. G. D. Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995).

13. Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997).

14. von Neumann, J. Beweis des Ergodensatzes und des H-theorems in der neuen mechanik. Z. Phys. 57, 20 (1929). The English translation is given in: Proof of the ergodic theorem and the H-theorem in quantum mechanics. Eur. Phys. J. H 35, 201–237 (2010).

15. Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).

16. Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).

17. Rigol, M. & Srednicki, M. Phys. Rev. Lett. 108, 110601 (2012).

18. Goldstein, S., Lebowitz, J. L., Tumulka, R. & Zanghì, N. Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006).

19. Popescu, S., Short, A. J. & Winter, A. Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754–758 (2006).

20. Imada, M. & Takahashi, M. Quantum transfer Monte Carlo method for finite temperature properties and quantum molecular dynamics method for dynamical correlation functions. J. Phys. Soc. Jpn 55, 3354–3361 (1986).

21. Jaklič, J. & Prelovšek, P. Lanczos method for the calculation of finite-temperature quantities in correlated systems. Phys. Rev. B 49, 5065–5068 (1994).

22. Hams, A. & De Raedt, H. Fast algorithm for finding the eigenvalue distribution of very large matrices. Phys. Rev. E 62, 4365–4377 (2000).

23. Sugiura, S. & Shimizu, A. Thermal pure quantum states at finite temperature. Phys. Rev. Lett. 108, 240401 (2012).

24. Sugiura, S. & Shimizu, A. Canonical thermal pure quantum state. Phys. Rev. Lett. 111, 010401 (2013).

25. Sugita, A. On the foundation of quantum statistical mechanics, RIMS Kokyuroku 1507, 147–159 (2006).

26. Sugita, A. On the basis of quantum statistical mechanics. Nonlinear Phenom. Complex Syst. 10, 192–195 (2007).

27. Reimann, P. Typicality for generalized microcanonical ensembles. Phys. Rev. Lett. 99, 160404 (2007).

28. Saito, K., Takesue, S. & Miyashita, S. System-size dependence of statistical behavior in quantum system. J. Phys. Soc. Jpn 65, 1243–1249 (1996).

29. Reimann, P. Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008).

30. Short, A. J. & Farrelly, T. C. Quantum equilibration in finite time. New J. Phys. 14, 013063 (2012).

31. Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).

32. Rigol, M. & Santos, L. F. Quantum chaos and thermalization in gapped systems. Phys. Rev. A 82, 011604 (2010).

33. Santos, L. F. & Rigol, M. Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization. Phys. Rev. E 81, 036206 (2010).

34. Santos, L. F. & Rigol, M. Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems. Phys. Rev. E 82, 031130 (2010).

35. Santos, L. F., Borgonovi, F. & Izrailev, F. M. Chaos and statistical relaxation in quantum systems of interacting particles. Phys. Rev. Lett. 108, 094102 (2012).

36. Santos, L. F., Borgonovi, F. & Izrailev, F. M. Onset of chaos and relaxation in isolated systems of interacting spins: energy shell approach. Phys. Rev. E 85, 036209 (2012).

37. Rigol, M. Breakdown of thermalization in finite one-dimensional systems. Phys. Rev. Lett. 103, 100403 (2009).

38. Rigol, M. Quantum quenches and thermalization in one-dimensional fermionic systems. Phys. Rev. A 80, 053607 (2009).

39. Biroli, G., Kollath, C. & L’áuchli, A. Effect of rare fluctuations on the thermalization of isolated quantum systems. Phys. Rev. Lett. 105, 250401 (2010).

40. Ikeda, T. N., Watanabe, Y. & Ueda, M. Finite-size scaling analysis of the eigenstate thermalization hypothesis in a one-dimensional interacting Bose gas. Phys. Rev. E 87, 012125 (2013).

41. Alba, V. Eigenstate thermalization hypothesis and integrability in quantum spin chains. Phys. Rev. B 91, 155123 (2015).

42. Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

43. Altman, E. & Vosk, R. Universal dynamics and renormalization in many-body-localized systems. Annu. Rev. Condens. Matter Phys. 6, 383–409 (2015).

44. Imbrie, J. Z., Ros, V. & Scardicchio, A. Local integrals of motion in many-body localized systems. Ann. Phys. 529, 1600278 (2017).

45. Srednicki, M. The approach to thermal equilibrium in quantized chaotic systems. J. Phys. A 32, 1163–1175 (1999).

46. Beugeling, W., Osessner, R. & Haque, M. Off-diagonal matrix elements of local operators in many-body quantum systems. Phys. Rev. E 91, 012144 (2015).

47. Hamazaki, R. & Ueda, M. Generalized Gibbs ensemble in a nonintegrable system with an extensive number of local symmetries. Phys. Rev. E 93, 032116 (2016).

48. Mondaini, R. & Rigol, M. Eigenstate thermalization in the two-dimensional transverse field Ising model. II. Off-diagonal matrix elements of observables. Phys. Rev. E 96, 012157 (2017).

49. Kim, H., Ikeda, T. N. & Huse, D. Testing whether all eigenstates obey the eigenstate thermalization hypothesis. Phys. Rev. E 90, 052105 (2014).

50. Iyoda, E., Kaneko, K. & Sagawa, T. Fluctuation theorem for many-body pure quantum states. Phys. Rev. Lett. 119, 100601 (2017).

51. Mori, T. Weak eigenstate thermalization with large deviation bound. Preprint at https://arxiv.org/abs/1609.09776 (2016).

52. Rogol, M. Fundamental asymmetry in quenches between integrable and nonintegrable systems. Phys. Rev. Lett. 116, 100601 (2016).

53. Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).

54. Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017).

55. Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

56. Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).

57. Blanes, S., Casas, F., Oteo, J. & Ros, J. The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009).

58. D’Alessio, L. & Polkovnikov, A. Many-body energy localization transition in periodically driven systems. Ann. Phys. 333, 19–33 (2013).

59. D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).

60. Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).

61. Ponte, P., Chandran, A., Papić, Z. & Abanin, D. A. Periodically driven ergodic and many-body localized quantum systems. Ann. Phys. 353, 196–204 (2015).

62. Russomanno, A., Silva, A. & Santoro, G. E. Periodic steady regime and interference in a periodically driven quantum system. Phys. Rev. Lett. 109, 257101 (2012).

63. Lazarides, A., Das, A. & Moessner, R. Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115, 030402 (2015).

64. Altman, E. Many-body localization and quantum thermalization. Nat. Phys. 14, 979–983 (2018).

65. Imbrie, J. Z. On many-body localization for quantum spin chains. J. Stat. Phys. 163, 998–1048 (2016).

66. Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).

67. Kondov, S. S., McGehee, W. R., Xu, W. & DeMarco, B. Disorder-induced localization in a strongly correlated atomic Hubbard gas. Phys. Rev. Lett. 114, 083002 (2015).

68. J-y, Choi et al. Exploring the many-body localization transition in two dimensions. Science 352, 1547–1552 (2016).

69. Goldstein, S., Lebowitz, J. L., Mastrodonato, C., Tumulka, R. & Zanghi, N. Phys. Rev. E 81, 011109 (2010).

70. Tasaki, H. The approach to thermal equilibrium and “thermodynamic normality” — an observation based on the works by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi in 2009, and by von Neumann in 1929. Preprint at https://arxiv.org/abs/1003.5424 (2010).

71. Reimann, P. Generalization of von Neumann’s approach to thermalization. Phys. Rev. Lett. 115, 010403 (2015).

72. Hamazaki, R. & Ueda, M. Atypicality of most few-body observables. Phys. Rev. Lett. 120, 080603 (2018).

73. Berges, J., Borsányi, S. & Wetterrich, C. Prethermalization. Phys. Rev. Lett. 93, 142002 (2004).

74. Rigol, M., Dunjko, V., Yurovsky, V. & Olshanii, M. Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007).

75. Berges, J., Rothkopf, A. & Schmidt, J. Nonthermal fixed points: effective weak coupling for strongly correlated systems far from equilibrium. Phys. Rev. Lett. 101, 041603 (2008).

76. Cassidy, A. C., Clark, C. W. & Rigol, M. Generalized thermalization in an integrable lattice system. Phys. Rev. Lett. 106, 140405 (2011).

77. Mallayya, K., Rigol, M. & De Roeck, W. Prethermalization and thermalization in isolated quantum systems. Phys. Rev. X 9, 21027 (2019).

78. Sotiriadis, S. & Calabrese, P. Validity of the GGE for quantum quenches from interacting to noninteracting models. J. Stat. Mech. P07024 (2014).

79. Pozsgay, B. et al. Correlations after quantum quenches in the XXZ spin chain: failure of the generalized Gibbs ensemble. Phys. Rev. Lett. 113, 117203 (2014).

80. Wouters, B. et al. Quenching the anisotropic Heisenberg chain: exact solution and generalized Gibbs ensemble predictions. Phys. Rev. Lett. 113, 117202 (2014).

81. Mierzejewski, M., Prelovsek, P. & Prosen, T. Identifying local and quasilocal conserved quantities in integrable systems. Phys. Rev. Lett. 114, 140601 (2015).

82. Ilievski, E., Medenjak, M. & Prosen, T. Quasilocal conserved operators in the isotropic Heisenberg spin-1/2 chain. Phys. Rev. Lett. 115, 120601 (2015).

83. Ilievski, E. et al. Complete generalized Gibbs ensenbles in an interacting theory. Phys. Rev. Lett. 115, 157201 (2015).

84. Kuwahara, T., Mori, T. & Saito, K. Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems. Ann. Phys. 367, 96–124 (2016).

85. Mori, T., Kuwahara, T. & Saito, K. Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems. Phys. Rev. Lett. 116, 120401 (2016).

86. Abanin, D. A., De Roeck, W., Ho, W. W. & Huveneers, F. Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017).

87. Abanin, D., De Roeck, W., Ho, W. W. & Huveneers, F. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Commun. Math. Phys. 354, 809–827 (2017).

88. Kollath, C., L’áuchli, A. M. & Altman, E. Quench dynamics and nonequilibrium phase diagram of the Bose–Hubbard model. Phys. Rev. Lett. 98, 180601 (2007).

89. Kaminishi, E., Mori, T., Ikeda, T. N. & Ueda, M. Entanglement pre-thermalization in a one-dimensional Bose gas. Nat. Phys. 11, 1050–1056 (2015).

90. Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963).

91. Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum Newtonas cradle. Nature 44, 900–903 (2006).

92. Wu, H. & Foot, C. Direct simulation of evaporative cooling. J. Phys. B 29, L321–L328 (1996).

93. Gring, M. et al. Relaxation and prethermalization in an isolated quantum system. Science 337, 1318–1322 (2012).

94. Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, J. Local emergence of thermal correlations in an isolated quantum many-body system. Nat. Phys. 9, 640–643 (2013).

95. Langen, T. et al. Experimental observation of a generalized Gibbs ensemble. Science 348, 207–211 (2015).

96. Jaynes, E. T. Information theory and statistical mechanics. II. Phys. Rev. 108, 171–190 (1957).

97. Cazalilla, M. Effect of suddenly turning on interactions in the Luttinger model. Phys. Rev. Lett. 97, 156403 (2006).

98. Caux, J. & Konik, R. Constructing the generalized Gibbs ensemble after a quantum quench. Phys. Rev. Lett. 109, 175301 (2012).

99. Tang, Y. et al. Thermalization near integrability in a dipolar quantum Newton’s cradle. Phys. Rev. X 8, 021030 (2018).

100. Giamarchi, T. Quantum Physics in One Dimension (Clarendon Press, 2003).

101. Orioli, A. P. et al. Relaxation of an isolated dipolar-interacting Rydberg quantum spin system. Phys. Rev. Lett. 120, 063601 (2018).

102. Rubio-Abadal, A. et al. Floquet prethermalization in a Bose–Hubbard system. Phys. Rev. X 10, 021044 (2020).

103. Bakr, W. S. et al. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009).

104. Bakr, W. S. et al. Probing the superfluid-to-Mott Insulator transition at the single-atom level. Science 329, 547–550 (2010).

105. Sherson, J. F. et al. Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 68–73 (2010).

106. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated stronly correlated one-dimensional Bose gas. Nat. Phys. 8, 325–330 (2012).

107. Lieb, E. H. & Robinson, D. W. The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972).

108. Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484–487 (2012).

109. Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794–800 (2016).

110. Ribzheimer, J. P. et al. Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions. Phys. Rev. Lett. 110, 205301 (2013).

111. Pal, A. & Huse, D. A. Many-body localization phase transition. Phys. Rev. B 82, 174411 (2010).

112. Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

113. Gomyi, I. V., Mirlin, A. D. & Polyakov, D. G. Interacting electrons in disordered wires: Anderson localization and low-T transport. Phys. Rev. Lett. 95, 206603 (2005).

114. Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).

115. Aleiner, I. L., Altshuler, B. L. & Shlyapnikov, G. V. A finite-temperature phase transition for disordered weakly interacting bosons in one dimension. Nat. Phys. 6, 900–904 (2010).

116. Aubry, S. & André, G. Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133–164 (1980).

117. Žnidarič, M., Prosen, T. & Prelovšek, P. Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008).

118. Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett. 109, 017202 (2012).

119. Vosk, R. & Altman, E. Many-body localization in one dimension as a dynamical renormalization group fixed point. Phys. Rev. Lett. 110, 260601 (2013).

120. Serbyn, M., Papić, Z. & Abanin, D. A. Universal slow growth of entanglement in interacting strongly disordered systems. Phys. Rev. Lett. 110, 260601 (2013).

121. Nanduri, A., Kim, H. & Huse, D. A. Entanglement spreading in a many-body localized system. Phys. Rev. B 90, 064201 (2014).

122. Bordia, P. et al. Coupling identical one-dimensional many-body localized systems. Phys. Rev. Lett. 116, 140401 (2016).

123. Lüschen, H. P. et al. Signature of many-body localization in a controlled open quantum system. Phys. Rev. X 7, 011034 (2017).

124. Bordia, P. et al. Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems. Phys. Rev. X 7, 041047 (2017).

125. Lüschen, H. P. et al. Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems. Phys. Rev. Lett. 119, 260401 (2017).

126. Rubio-Abadal, A. et al. Many-body delocalization in the presence of a quantum bath. Phys. Rev. X 9, 041014 (2019).

127. Lukin, A. et al. Probing entanglement in a many-body-localized system. Science 364, 256–260 (2019).

128. Kiefer-Emmanouilidis, M., Unayan, R., Fleischauer, M. & Sirker, J. Evidence for unbounded growth of the number entropy in many-body localized phases. Phys. Rev. Lett. 124, 243601 (2020).

129. Bordia, P., Lüschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).

130. Nowak, B., Sexty, D. & Gasenzer, T. Superfluid turbulence: nonthermal fixed point in an ultracold Bose gas. Phys. Rev. B 84, 020506(R) (2011).

131. PiñeiroOrioli, A., Boguslavski, K. & Berges, J. Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points. Phys. Rev. D 92, 025041 (2015).

132. Navon, N., Gaunt, A. L., Smith, R. P. & Hadzibabic, Z. Emergence of a turbulent cascade in a quantum gas. Nature 539, 72–75 (2016).

133. Prüfer, M. et al. Observation of universal dynamics in a spinor bose gas far from equilibrium. Nature 563, 217–221 (2018).

134. Erne, S., Bücker, R., Gasenzer, T., Berges, J. & Schmiedmayer, J. Universal dynamics in an isolated one-dimensional bose gas far from equilibrium. Nature 563, 225–229 (2018).

135. Gaunt, A. L., Fletcher, R. J., Smith, R. P. & Hadzibabic, Z. A superheated Bose-condensed gas. Nat. Phys. 9, 271–274 (2013).

136. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

137. Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum many-body scars. Nat. Phys. 14, 745–749 (2018).

138. Choi, S. et al. Emergent SU(2) dynamics and perfect quantum many-body scars. Phys. Rev. Lett. 122, 220603 (2019).

139. Rauer, B. et al. Recurrences in an isolated quantum many-body system. Science 360, 307–310 (2018).

140. Saint-Jalm, R. et al. Dynamical symmetry and breathers in a two-dimensional Bose gas. Phys. Rev. X 9, 021035 (2019).

## Acknowledgements

I acknowledge my present and former group members from whom I learned numerous things concerning the subjects described in this article. Special thanks are due to T. Mori, R. Hamazaki and Z. Gong who are always willing to share their expertise with me. This work was supported by KAKENHI grant number JP18H01145.

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Ueda, M. Quantum equilibration, thermalization and prethermalization in ultracold atoms. Nat Rev Phys 2, 669–681 (2020). https://doi.org/10.1038/s42254-020-0237-x

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• Alec Cao