Fermion production at the boundary of an expanding universe: a cold-atom gravitational analogue

Carlos Fulgado-Claudio, Jose M. Sánchez Velázquez, and Alejandro Bermudez

Instituto de Física Teórica, UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain.

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Abstract

We study the phenomenon of cosmological particle production of Dirac fermions in a Friedmann-Robertson-Walker spacetime, focusing on a $(1+1)-$dimensional case in which the evolution of the scale factor is set by the equations of Jackiw-Teitelboim gravity. As a first step towards a quantum simulation of this phenomenon, we consider two possible lattice regularizations, which allow us to explore the interplay of particle production and topological phenomena in spacetimes with a boundary. In particular, for a Wilson-type discretization of the Dirac field, the asymptotic Minkowski vacua connected by the intermediate expansion correspond to symmetry-protected topological groundstates, and have a boundary manifestation in the form of zero-modes exponentially localized to the spatial boundaries. We show that particle production can also populate these zero modes, which contrasts with the situation with a naïve-fermion discretization, in which conformal zero-mass fields do not allow for particle production. We present a scheme for the quantum simulation of this gravitational analogue by means of ultra-cold atoms in Raman optical lattices, which require real-time control of the Raman-beam detuning according to the scale factor of the simulated spacetime, as well as band-mapping measurements.

There is a striking phenomenon occurring for quantum fields in curved spacetimes, such as rapidly expanding geometries, which is gravitational particle production. Recently, there have been several experimental proposals within the realm of quantum simulations that can implement the dynamics of cosmological scenarios, paving thus the way for getting insights on several aspects of this phenomenon. We propose a new experimental setup to study fermion production in a cold-atom gravitational analogue.

Among the possibilities that this viewpoint enables, we focus on the interactions between cosmological particle production and topological phases, which are phases of matter governed by topological invariants. We consider a universe with reduced dimensionality, where the presence of these phases is manifested in the appearance of zero-modes exponentially localized to the spatial boundaries of the universe. This allows us to study how these modes interact with the expanding background and to tackle the differences with respect to the creation of propagating particles.

This framework allows for the study of exotic phases of matter within cosmological contexts, and it opens the way for the exploration of more complex models. For instance, interactions can be easily implemented, allowing thus for the study of non-perturbative phenomena. Also, it would be of great interest to generalise the experimental setup to higher-dimensional universes, where different topological phases would appear. These features can be easily implemented in our experimental scheme.

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► References

[1] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995).
https:/​/​doi.org/​10.1201/​9780429503559

[2] E. Fradkin, Field Theories of Condensed Matter Physics, 2nd ed. (Cambridge University Press, 2013).
https:/​/​doi.org/​10.1017/​CBO9781139015509

[3] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).
https:/​/​doi.org/​10.1017/​CBO9780511973765

[4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.109

[5] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
https:/​/​doi.org/​10.1103/​RevModPhys.90.015001

[6] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
https:/​/​doi.org/​10.1103/​RevModPhys.83.1057

[7] P. W. Anderson, Science 177, 393 (1972).
https:/​/​doi.org/​10.1126/​science.177.4047.393

[8] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).
https:/​/​doi.org/​10.1103/​PhysRevLett.46.1351

[9] S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Cambridge University Press, 2019).
https:/​/​doi.org/​10.1017/​9781108770385

[10] C. Barceló, S. Liberati, and M. Visser, Living Reviews in Relativity 8, 12 (2005).
https:/​/​doi.org/​10.12942/​lrr-2005-12

[11] M. J. Jacquet, S. Weinfurtner, and F. König, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, 20190239 (2020).
https:/​/​doi.org/​10.1098/​rsta.2019.0239

[12] L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2009).
https:/​/​doi.org/​10.1017/​CBO9780511813924

[13] S. W. Hawking, Nature 248, 30 (1974).
https:/​/​doi.org/​10.1038/​248030a0

[14] A. H. Guth, Phys. Rev. D 23, 347 (1981).
https:/​/​doi.org/​10.1103/​PhysRevD.23.347

[15] F. Belgiorno, S. L. Cacciatori, G. Ortenzi, L. Rizzi, V. Gorini, and D. Faccio, Phys. Rev. D 83, 024015 (2011).
https:/​/​doi.org/​10.1103/​PhysRevD.83.024015

[16] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, Science 319, 1367 (2008).
https:/​/​doi.org/​10.1126/​science.1153625

[17] F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Faccio, Phys. Rev. Lett. 105, 203901 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.203901

[18] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. Leonhardt, Phys. Rev. Lett. 122, 010404 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.010404

[19] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.85.4643

[20] O. Lahav, A. Itah, A. Blumkin, C. Gordon, S. Rinott, A. Zayats, and J. Steinhauer, Phys. Rev. Lett. 105, 240401 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.240401

[21] J. Steinhauer, Nature Physics 12, 959 (2016).
https:/​/​doi.org/​10.1038/​nphys3863

[22] V. I. Kolobov, K. Golubkov, J. R. Muñoz de Nova, and J. Steinhauer, Nature Physics 17, 362 (2021).
https:/​/​doi.org/​10.1038/​s41567-020-01076-0

[23] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
https:/​/​doi.org/​10.1103/​PhysRevD.14.870

[24] J. Hu, L. Feng, Z. Zhang, and C. Chin, Nature Physics 15, 785 (2019).
https:/​/​doi.org/​10.1038/​s41567-019-0537-1

[25] L. Parker, Phys. Rev. Lett. 21, 562 (1968).
https:/​/​doi.org/​10.1103/​PhysRevLett.21.562

[26] L. Parker, Phys. Rev. 183, 1057 (1969).
https:/​/​doi.org/​10.1103/​PhysRev.183.1057

[27] C. Barceló, S. Liberati, and M. Visser, Phys. Rev. A 68, 053613 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.68.053613

[28] P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69, 033602 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.033602

[29] C. Fey, T. Schaetz, and R. Schützhold, Phys. Rev. A 98, 033407 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.033407

[30] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, Phys. Rev. X 8, 021021 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.021021

[31] M. Wittemer, F. Hakelberg, P. Kiefer, J.-P. Schröder, C. Fey, R. Schützhold, U. Warring, and T. Schaetz, Phys. Rev. Lett. 123, 180502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.180502

[32] C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, Á. Parra-López, M. Tolosa-Simeón, N. Sánchez-Kuntz, T. Haas, H. Strobel, S. Floerchinger, and M. K. Oberthaler, Nature 611, 260 (2022).
https:/​/​doi.org/​10.1038/​s41586-022-05313-9

[33] M. Tolosa-Simeón, A. Parra-López, N. Sánchez-Kuntz, T. Haas, C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, H. Strobel, M. K. Oberthaler, and S. Floerchinger, Phys. Rev. A 106, 033313 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.106.033313

[34] N. Sanchez-Kuntz, A. Parra-López, M. Tolosa-Simeón, T. Haas, and S. Floerchinger, Physical Review D 105 (2022), 10.1103/​physrevd.105.105020.
https:/​/​doi.org/​10.1103/​physrevd.105.105020

[35] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008a).
https:/​/​doi.org/​10.1103/​RevModPhys.80.885

[36] L. Parker, Phys. Rev. D 3, 346 (1971).
https:/​/​doi.org/​10.1103/​PhysRevD.3.346

[37] T. Kaluza, International Journal of Modern Physics D 27, 1870001 (2018).
https:/​/​doi.org/​10.1142/​S0218271818700017

[38] O. Klein, Zeitschrift für Physik 37, 895 (1926).
https:/​/​doi.org/​10.1007/​BF01397481

[39] F. Khanna, A. Malbouisson, J. Malbouisson, and A. Santana, Physics Reports 539, 135 (2014).
https:/​/​doi.org/​10.1016/​j.physrep.2014.02.002

[40] R. M. Wald, Living Reviews in Relativity 4, 6 (2001).
https:/​/​doi.org/​10.12942/​lrr-2001-6

[41] H. B. G. Casimir, Proc. Akad. Wet. Amsterdam 51, 793 (1948).

[42] G. T. Moore, Journal of Mathematical Physics 11, 2679 (1970).
https:/​/​doi.org/​10.1063/​1.1665432

[43] E. Witten, Communications in Mathematical Physics 121, 351 (1989).

[44] S. Elitzur, G. Moore, A. Schwimmer, and N. Seiberg, Nuclear Physics B 326, 108 (1989).
https:/​/​doi.org/​10.1016/​0550-3213(89)90436-7

[45] A. Balachandran, L. Chándal, and E. Ercolessi, International Journal of Modern Physics A 10, 1969 (1995).
https:/​/​doi.org/​10.1142/​S0217751X95000966

[46] M. F. Golterman, K. Jansen, and D. B. Kaplan, Physics Letters B 301, 219 (1993).
https:/​/​doi.org/​10.1016/​0370-2693(93)90692-B

[47] C. Callan and J. Harvey, Nuclear Physics B 250, 427 (1985).
https:/​/​doi.org/​10.1016/​0550-3213(85)90489-4

[48] D. B. Kaplan, in Les Houches Summer School: Session 93: Modern perspectives in lattice QCD: Quantum field theory and high performance computing (2009) pp. 223–272, arXiv:0912.2560 [hep-lat].
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199691609.001.0001
arXiv:0912.2560

[49] D. B. Kaplan, Physics Letters B 288, 342 (1992).
https:/​/​doi.org/​10.1016/​0370-2693(92)91112-M

[50] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
https:/​/​doi.org/​10.1007/​bf02650179

[51] D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).
https:/​/​doi.org/​10.1103/​PhysRevD.10.3235

[52] J. D. Brown, Lower Dimensional Gravity (World Scientific, 1988).
https:/​/​doi.org/​10.1142/​0622

[53] L. H. Ford, Reports on Progress in Physics 84, 116901 (2021).
https:/​/​doi.org/​10.1088/​1361-6633/​ac1b23

[54] V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, 2007).
https:/​/​doi.org/​10.1017/​CBO9780511809149

[55] J. Preskill, ``Quantum field theory in curved spacetime,'' (1990).
http:/​/​theory.caltech.edu/​~preskill/​notes.html

[56] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984).
https:/​/​doi.org/​10.1017/​CBO9780511622632

[57] T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360, 117 (1978).
https:/​/​doi.org/​10.1098/​rspa.1978.0060

[58] J. F. Barbero G., A. Ferreiro, J. Navarro-Salas, and E. J. S. Villaseñor, Phys. Rev. D 98, 025016 (2018).
https:/​/​doi.org/​10.1103/​PhysRevD.98.025016

[59] J. G. Valatin, Il Nuovo Cimento 7, 843 (1958).
https:/​/​doi.org/​10.1007/​BF02745589

[60] N. N. Bogolyubov, Il Nuovo Cimento 7, 794 (1958).
https:/​/​doi.org/​10.1007/​BF02745585

[61] A. E. Sikkema and R. B. Mann, Classical and Quantum Gravity 8, 219 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​1/​022

[62] P. R. Anderson, E. Mottola, and D. H. Sanders, Phys. Rev. D 97, 065016 (2018), arXiv:1712.04522 [gr-qc].
https:/​/​doi.org/​10.1103/​PhysRevD.97.065016
arXiv:1712.04522

[63] A. Duncan, Phys. Rev. D 17, 964 (1978).
https:/​/​doi.org/​10.1103/​PhysRevD.17.964

[64] I. Fuentes, R. B. Mann, E. Martín-Martínez, and S. Moradi, Phys. Rev. D 82, 045030 (2010).
https:/​/​doi.org/​10.1103/​PhysRevD.82.045030

[65] E. Martín-Martínez and N. C. Menicucci, Classical and Quantum Gravity 29, 224003 (2012).
https:/​/​doi.org/​10.1088/​0264-9381/​29/​22/​224003

[66] A. Grib, S. Mamayev, V. Mostepanenko, and V. Mostepanenko, Vacuum Quantum Effects in Strong Fields (Friedmann Laboratory Pub., 1994).
https:/​/​books.google.es/​books?id=azBdcgAACAAJ

[67] J. Haro and E. Elizalde, Journal of Physics A: Mathematical and Theoretical 41, 372003 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​37/​372003

[68] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Quantum physics, quantum information and quantum computation (Cambridge University Press, 2020).
https:/​/​doi.org/​10.1017/​9781108587280

[69] D. J. H. Chung, L. L. Everett, H. Yoo, and P. Zhou, Phys. Lett. B 712, 147 (2012), arXiv:1109.2524 [astro-ph.CO].
https:/​/​doi.org/​10.1016/​j.physletb.2012.04.066
arXiv:1109.2524

[70] M. Abramowitz, I. A. Stegun, and R. H. Romer, American Journal of Physics 56, 958 (1988).
https:/​/​doi.org/​10.1119/​1.15378

[71] Y. Ema, K. Nakayama, and Y. Tang, Journal of High Energy Physics 2019, 60 (2019).
https:/​/​doi.org/​10.1007/​JHEP07(2019)060

[72] D. E. Borrajo Gutiérrez, J. A. R. Cembranos, L. J. Garay, and J. M. Sánchez Velázquez, Journal of High Energy Physics 2020, 69 (2020).
https:/​/​doi.org/​10.1007/​JHEP09(2020)069

[73] V. M. Villalba, Phys. Rev. D 52, 3742 (1995), arXiv:hep-th/​9507021.
https:/​/​doi.org/​10.1103/​PhysRevD.52.3742
arXiv:hep-th/9507021

[74] K. G. Wilson and J. Kogut, Physics Reports 12, 75 (1974).
https:/​/​doi.org/​10.1016/​0370-1573(74)90023-4

[75] K. G. Wilson, Phys. Rev. D 10, 2445 (1974a).
https:/​/​doi.org/​10.1103/​PhysRevD.10.2445

[76] C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice, Vol. 788 (Springer, Berlin, 2010).
https:/​/​doi.org/​10.1007/​978-3-642-01850-3

[77] H. J. Rothe, Lattice Gauge Theories : An Introduction (Fourth Edition), Vol. 43 (World Scientific Publishing Company, 2012).
https:/​/​doi.org/​10.1142/​8229

[78] K. G. Wilson, in New Phenomena in Subnuclear Physics (Springer US, 1977) pp. 69–142.
https:/​/​doi.org/​10.1007/​978-1-4613-4208-3_6

[79] H. So, Progress of Theoretical Physics 73, 528 (1985).
https:/​/​doi.org/​10.1143/​PTP.73.528

[80] A. Bermudez, L. Mazza, M. Rizzi, N. Goldman, M. Lewenstein, and M. A. Martin-Delgado, Phys. Rev. Lett. 105, 190404 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.190404

[81] D. B. Kaplan and S. Sun, Phys. Rev. Lett. 108, 181807 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.181807

[82] J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, Phys. Rev. X 7, 031057 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.031057

[83] A. Bermudez, E. Tirrito, M. Rizzi, M. Lewenstein, and S. Hands, Annals of Physics 399, 149 (2018).
https:/​/​doi.org/​10.1016/​j.aop.2018.10.007

[84] Y. Kuno, Phys. Rev. B 99, 064105 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.064105

[85] E. Tirrito, M. Rizzi, G. Sierra, M. Lewenstein, and A. Bermudez, Phys. Rev. B 99, 125106 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.125106

[86] S. Sen, Phys. Rev. D 102, 094520 (2020).
https:/​/​doi.org/​10.1103/​PhysRevD.102.094520

[87] D. B. Kaplan and S. Sen, ``Index theorems, generalized hall currents and topology for gapless defect fermions,'' (2021).
https:/​/​doi.org/​10.48550/​ARXIV.2112.06954

[88] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annual Review of Condensed Matter Physics 7, 301 (2016).
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031115-011417

[89] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
https:/​/​doi.org/​10.1103/​PhysRevLett.61.2015

[90] F. D. M. Haldane, Rev. Mod. Phys. 89, 040502 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.040502

[91] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).
https:/​/​doi.org/​10.1103/​PhysRevB.78.195424

[92] L. D. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937).
https:/​/​doi.org/​10.1016/​B978-0-08-010586-4.50034-1

[93] X.-G. Wen, Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons (Oxford University Press, Oxford, 2007).
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199227259.001.0001

[94] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
https:/​/​doi.org/​10.1103/​PhysRevLett.49.405

[95] M. Nakahara, Geometry, Topology and Physics (CRC Press, 2017).
https:/​/​doi.org/​10.1201/​9781315275826

[96] T. Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015).
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031214-014740

[97] A. Kitaev, AIP Conference Proceedings 1134, 22 (2009).
https:/​/​doi.org/​10.1063/​1.3149495

[98] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
https:/​/​doi.org/​10.1103/​RevModPhys.88.035005

[99] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New Journal of Physics 12, 065010 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​6/​065010

[100] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
https:/​/​doi.org/​10.1103/​PhysRevLett.42.1698

[101] J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators, Vol. 919 (2016).
https:/​/​doi.org/​10.1007/​978-3-319-25607-8

[102] R. Shankar, ``Topological insulators – a review,'' (2018).
https:/​/​doi.org/​10.48550/​ARXIV.1804.06471

[103] J. Zak, Phys. Rev. Lett. 62, 2747 (1989).
https:/​/​doi.org/​10.1103/​PhysRevLett.62.2747

[104] M. V. Berry, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).
https:/​/​doi.org/​10.1098/​rspa.1984.0023

[105] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
https:/​/​doi.org/​10.1103/​RevModPhys.82.1959

[106] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).
https:/​/​doi.org/​10.1103/​PhysRevD.13.3398

[107] G. Magnifico, D. Vodola, E. Ercolessi, S. P. Kumar, M. Müller, and A. Bermudez, Phys. Rev. D 99, 014503 (2019a).
https:/​/​doi.org/​10.1103/​PhysRevD.99.014503

[108] G. Magnifico, D. Vodola, E. Ercolessi, S. P. Kumar, M. Müller, and A. Bermudez, Phys. Rev. B 100, 115152 (2019b).
https:/​/​doi.org/​10.1103/​PhysRevB.100.115152

[109] J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).
https:/​/​doi.org/​10.1103/​PhysRevD.11.395

[110] L. Susskind, Phys. Rev. D 16, 3031 (1977).
https:/​/​doi.org/​10.1103/​PhysRevD.16.3031

[111] M. Creutz, Phys. Rev. Lett. 83, 2636 (1999a).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.2636

[112] M. Creutz, Rev. Mod. Phys. 73, 119 (2001).
https:/​/​doi.org/​10.1103/​RevModPhys.73.119

[113] L. Mazza, A. Bermudez, N. Goldman, M. Rizzi, M. A. Martin-Delgado, and M. Lewenstein, New Journal of Physics 14, 015007 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​1/​015007

[114] C. G. Velasco and B. Paredes, (2019), arXiv:1907.11460 [cond-mat.quant-gas].
arXiv:1907.11460

[115] J. Zurita, C. Creffield, and G. Platero, Quantum 5, 591 (2021).
https:/​/​doi.org/​10.22331/​q-2021-11-25-591

[116] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Phys. Rev. B 86, 155140 (2012).
https:/​/​doi.org/​10.1103/​PhysRevB.86.155140

[117] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Phys. Rev. Lett. 112, 130401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.130401

[118] A. Bermudez, D. Patanè, L. Amico, and M. A. Martin-Delgado, Phys. Rev. Lett. 102, 135702 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.135702

[119] R. Jafari, H. Johannesson, A. Langari, and M. A. Martin-Delgado, Phys. Rev. B 99, 054302 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.054302

[120] N. Sun and L.-K. Lim, Phys. Rev. B 96, 035139 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.96.035139

[121] A. Haller, M. Rizzi, and M. Filippone, Phys. Rev. Research 2, 023058 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023058

[122] S. M. Chan, B. Grémaud, and G. G. Batrouni, Phys. Rev. B 105, 024502 (2022).
https:/​/​doi.org/​10.1103/​PhysRevB.105.024502

[123] T. Orito, Y. Kuno, and I. Ichinose, Phys. Rev. B 105, 094201 (2022).
https:/​/​doi.org/​10.1103/​PhysRevB.105.094201

[124] T. Orito, Y. Kuno, and I. Ichinose, Phys. Rev. B 103, L060301 (2021a).
https:/​/​doi.org/​10.1103/​PhysRevB.103.L060301

[125] T. Orito, Y. Kuno, and I. Ichinose, Phys. Rev. B 104, 094202 (2021b).
https:/​/​doi.org/​10.1103/​PhysRevB.104.094202

[126] G.-Q. Zhang, L.-Z. Tang, L.-F. Zhang, D.-W. Zhang, and S.-L. Zhu, Phys. Rev. B 104, L161118 (2021).
https:/​/​doi.org/​10.1103/​PhysRevB.104.L161118

[127] D. González-Cuadra, L. Tagliacozzo, M. Lewenstein, and A. Bermudez, Phys. Rev. X 10, 041007 (2020).
https:/​/​doi.org/​10.1103/​PhysRevX.10.041007

[128] E. Tirrito, M. Lewenstein, and A. Bermudez, Phys. Rev. B 106, 045147 (2022).
https:/​/​doi.org/​10.1103/​PhysRevB.106.045147

[129] K. G. Wilson, Phys. Rev. D 10, 2445 (1974b).
https:/​/​doi.org/​10.1103/​PhysRevD.10.2445

[130] T. Kibble, Physics Reports 67, 183 (1980).
https:/​/​doi.org/​10.1016/​0370-1573(80)90091-5

[131] W. Zurek, Physics Reports 276, 177 (1996).
https:/​/​doi.org/​10.1016/​S0370-1573(96)00009-9

[132] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.105701

[133] M. Creutz, Phys. Rev. Lett. 83, 2636 (1999b).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.2636

[134] I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics 8, 267 (2012).
https:/​/​doi.org/​10.1038/​nphys2259

[135] N. Goldman, J. C. Budich, and P. Zoller, Nature Physics 12, 639 (2016).
https:/​/​doi.org/​10.1038/​nphys3803

[136] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen, Advances in Physics 56, 243 (2007).
https:/​/​doi.org/​10.1080/​00018730701223200

[137] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, New Journal of Physics 13, 035002 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​3/​035002

[138] J. Minář and B. Grémaud, Journal of Physics A: Mathematical and Theoretical 48, 165001 (2015).
https:/​/​doi.org/​10.1088/​1751-8113/​48/​16/​165001

[139] J. Rodríguez-Laguna, L. Tarruell, M. Lewenstein, and A. Celi, Phys. Rev. A 95, 013627 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.013627

[140] A. Celi, The European Physical Journal Special Topics 226, 2729 (2017).
https:/​/​doi.org/​10.1140/​epjst/​e2016-60390-y

[141] A. Kosior, M. Lewenstein, and A. Celi, SciPost Phys. 5, 061 (2018).
https:/​/​doi.org/​10.21468/​SciPostPhys.5.6.061

[142] U.-J. Wiese, Annalen der Physik 525, 777 (2013).
https:/​/​doi.org/​10.1002/​andp.201300104

[143] E. Zohar, J. I. Cirac, and B. Reznik, Reports on Progress in Physics 79, 014401 (2015).
https:/​/​doi.org/​10.1088/​0034-4885/​79/​1/​014401

[144] M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, The European Physical Journal D 74, 165 (2020).
https:/​/​doi.org/​10.1140/​epjd/​e2020-100571-8

[145] M. C. Bañuls and K. Cichy, Reports on Progress in Physics 83, 024401 (2020).
https:/​/​doi.org/​10.1088/​1361-6633/​ab6311

[146] M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, A. Dauphin, D. González-Cuadra, P. R. Grzybowski, S. Hands, F. Jendrzejewski, J. Jünemann, G. Juzeliūnas, V. Kasper, A. Piga, S.-J. Ran, M. Rizzi, G. Sierra, L. Tagliacozzo, E. Tirrito, T. V. Zache, J. Zakrzewski, E. Zohar, and M. Lewenstein, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380, 20210064 (2022).
https:/​/​doi.org/​10.1098/​rsta.2021.0064

[147] N. Klco, A. Roggero, and M. J. Savage, Reports on Progress in Physics 85, 064301 (2022).
https:/​/​doi.org/​10.1088/​1361-6633/​ac58a4

[148] L. Ziegler, E. Tirrito, M. Lewenstein, S. Hands, and A. Bermudez, Phys. Rev. Research 4, L042012 (2022).
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.L042012

[149] L. Ziegler, E. Tirrito, M. Lewenstein, S. Hands, and A. Bermudez, Annals of Physics 439, 168763 (2022), arXiv:2111.04485.
https:/​/​doi.org/​10.1016/​j.aop.2022.168763
arXiv:2111.04485

[150] L. Zhang and X.-J. Liu, in Synthetic Spin-Orbit Coupling in Cold Atoms (WORLD SCIENTIFIC, 2018) pp. 1–87.
https:/​/​doi.org/​10.1142/​9789813272538_0001

[151] X.-J. Liu, Z.-X. Liu, and M. Cheng, Phys. Rev. Lett. 110, 076401 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.076401

[152] X.-J. Liu, K. T. Law, and T. K. Ng, Phys. Rev. Lett. 112, 086401 (2014a).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.086401

[153] X.-J. Liu, K. T. Law, and T. K. Ng, Phys. Rev. Lett. 113, 059901 (2014b).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.059901

[154] M. A. Cazalilla and A. M. Rey, Reports on Progress in Physics 77, 124401 (2014).
https:/​/​doi.org/​10.1088/​0034-4885/​77/​12/​124401

[155] M.-C. Liang, Y.-D. Wei, L. Zhang, X.-J. Wang, H. Zhang, W.-W. Wang, W. Qi, X.-J. Liu, and X. Zhang, Phys. Rev. Res. 5, L012006 (2023).
https:/​/​doi.org/​10.1103/​PhysRevResearch.5.L012006

[156] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 354, 83 (2016).
https:/​/​doi.org/​10.1126/​science.aaf6689

[157] W. Sun, B.-Z. Wang, X.-T. Xu, C.-R. Yi, L. Zhang, Z. Wu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 150401 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.150401

[158] B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S. Zhang, X.-J. Liu, and G.-B. Jo, Science Advances 4, eaao4748 (2018).
https:/​/​doi.org/​10.1126/​sciadv.aao4748

[159] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
https:/​/​doi.org/​10.1103/​PhysRevLett.81.3108

[160] W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 89, 220407 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.220407

[161] D. Jaksch and P. Zoller, New Journal of Physics 5, 56 (2003).
https:/​/​doi.org/​10.1088/​1367-2630/​5/​1/​356

[162] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.76.4508

[163] X. Huang and L. Parker, Phys. Rev. D 79, 024020 (2009).
https:/​/​doi.org/​10.1103/​PhysRevD.79.024020

[164] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008b).
https:/​/​doi.org/​10.1103/​revmodphys.80.885

[165] W. Ketterle and M. W. Zwierlein, La Rivista del Nuovo Cimento 31, 247–422 (2008).
https:/​/​doi.org/​10.1393/​ncr/​i2008-10033-1

[166] M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, and T. Esslinger, 87, 160405 (2001), arXiv:cond-mat/​0105105 [cond-mat.soft].
https:/​/​doi.org/​10.1103/​PhysRevLett.87.160405
arXiv:cond-mat/0105105

[167] M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger, Physical Review Letters 94 (2005), 10.1103/​physrevlett.94.080403.
https:/​/​doi.org/​10.1103/​physrevlett.94.080403

[168] A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and P. S. Jessen, 74, 1542 (1995).
https:/​/​doi.org/​10.1103/​PhysRevLett.74.1542

[169] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, 483, 302 (2012), arXiv:1111.5020 [cond-mat.quant-gas].
https:/​/​doi.org/​10.1038/​nature10871
arXiv:1111.5020

[170] E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos, and J. J. Garcia-Ripoll, Phys. Rev. Lett. 107, 235301 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.235301

[171] N. R. Cooper, J. Dalibard, and I. B. Spielman, Reviews of Modern Physics 91, 015005 (2019), arXiv:1803.00249 [cond-mat.quant-gas].
https:/​/​doi.org/​10.1103/​RevModPhys.91.015005
arXiv:1803.00249

[172] S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, 1995).
https:/​/​doi.org/​10.1017/​CBO9781139644167

[173] D. Z. Freedman and A. Van Proeyen, Supergravity (Cambridge University Press, 2012).
https:/​/​doi.org/​10.1017/​CBO9781139026833

[174] T. W. B. Kibble, Journal of Mathematical Physics 2, 212 (1961).
https:/​/​doi.org/​10.1063/​1.1703702

[175] I. L. Shapiro, Universe 8 (2022), 10.3390/​universe8110586.
https:/​/​doi.org/​10.3390/​universe8110586

[176] C. Teitelboim, Phys. Lett. B 126, 41 (1983).
https:/​/​doi.org/​10.1016/​0370-2693(83)90012-6

[177] R. Jackiw, Nucl. Phys. B 252, 343 (1985).
https:/​/​doi.org/​10.1016/​0550-3213(85)90448-1

[178] R. B. Mann, A. Shiekh, and L. Tarasov, Nucl. Phys. B 341, 134 (1990).
https:/​/​doi.org/​10.1016/​0550-3213(90)90265-F

[179] C. Sabín, New Journal of Physics 20, 053028 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aac0db

[180] J. F. García and C. Sabín, Phys. Rev. D 99, 025008 (2019).
https:/​/​doi.org/​10.1103/​PhysRevD.99.025008

[181] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, seventh ed. (Elsevier/​Academic Press, Amsterdam, 2007) pp. xlviii+1171, translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).

[182] A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Gen. Rel. Grav. 7, 535 (1976).
https:/​/​doi.org/​10.1007/​BF00766413

[183] Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Phys. Rev. D 51, 5438 (1995), arXiv:hep-ph/​9407247.
https:/​/​doi.org/​10.1103/​PhysRevD.51.5438
arXiv:hep-ph/9407247

[184] K. Svozil, Phys. Rev. Lett. 65, 3341 (1990).
https:/​/​doi.org/​10.1103/​PhysRevLett.65.3341

[185] M. Ban, J. Opt. Soc. Am. B 10, 1347 (1993).
https:/​/​doi.org/​10.1364/​JOSAB.10.001347

Cited by

[1] Vladimir Mostepanenko, "Prediction of the Expansion of the Universe Made by Alexander Friedmann and the Effect of Particle Creation in Cosmology", Universe 10 2, 84 (2024).

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