Finding zeros of the Riemann zeta function by periodic driving of cold atoms
| dc.contributor.author | Creffield, Charles | |
| dc.contributor.author | Sierra, G. | |
| dc.date.accessioned | 2023-06-18T06:46:21Z | |
| dc.date.available | 2023-06-18T06:46:21Z | |
| dc.date.issued | 2015-06-08 | |
| dc.description | ©2015 American Physical Society. The authors thank Michael Berry for stimulating discussions. C.E.C. was supported by the Spanish MINECO through Grants No. FIS2010-21372 and No. FIS2013-41716-P, and G.S. by Grant No. FIS2012-33642, QUITEMAD, and the Severo Ochoa Programme under Grant SEV-2012-0249. | |
| dc.description.abstract | The Riemann hypothesis, which states that the nontrivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. We propose here an approach to finding a physical system to study the Riemann zeros, which is based on applying a time-periodic driving field. This driving allows us to tune the quasienergies of the system (the analog of the eigenenergies for static systems), so that they are directly governed by the zeta function. We further show by numerical simulations that this allows the Riemann zeros to be measured in currently accessible cold- atom experiments. | |
| dc.description.department | Depto. de Física de Materiales | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | MINECO, Spain | |
| dc.description.sponsorship | Severo Ochoa Programme | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/32916 | |
| dc.identifier.doi | 10.1103/PhysRevA.91.063608 | |
| dc.identifier.issn | 1050-2947 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1103/PhysRevA.91.063608 | |
| dc.identifier.relatedurl | http://journals.aps.org | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/24123 | |
| dc.issue.number | 6 | |
| dc.journal.title | Physical review A | |
| dc.language.iso | eng | |
| dc.publisher | American Physical Society | |
| dc.relation.projectID | FIS2010-21372 | |
| dc.relation.projectID | FIS2013-41716-P | |
| dc.relation.projectID | FIS2012-33642 | |
| dc.relation.projectID | QUITEMAD | |
| dc.relation.projectID | SEV-2012-0249 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 538.9 | |
| dc.subject.keyword | Optics | |
| dc.subject.keyword | Physics | |
| dc.subject.keyword | Atomic | |
| dc.subject.keyword | Molecular & Chemical | |
| dc.subject.ucm | Física de materiales | |
| dc.subject.ucm | Física del estado sólido | |
| dc.subject.unesco | 2211 Física del Estado Sólido | |
| dc.title | Finding zeros of the Riemann zeta function by periodic driving of cold atoms | |
| dc.type | journal article | |
| dc.volume.number | 91 | |
| dcterms.references | [1] G. Sierra, A Physics Pathway to the Riemann Hypothesis, in Mathematical Physics and Field theory. Julio Abad, in Memoriam., edited by M. Asorey, J. V. Garcia-Esteve, M. F. Rañada, and J. Sesma (University of Zaragoza Press, Zaragoza, 2009), pp. 383–390; G. Sierra, arXiv:1012.4264. [2] D. Schumayer and D. A. W. Hutchinson, Rev. Mod. Phys. 83, 307 (2011). [3] H. L. Montgomery, Proc. Symp. Pure Math. 24, 181 (1973). [4] A. M. Odlyzko, Math. Comput. 48, 273 (1987). [5] M. V. Berry, in Quantum Chaos and Statistical Nuclear Physics, edited by T. H. Seligman and H. Nishioka, Springer Lecture Notes in Physics Vol. 263, p. 1 (Springer, New York, 1986). [6] J. P. Keating, in Supersymmetry and Trace Formulae: Chaos and Disorder, edited by I. V. erner et al.(Kluwer Academic/Plenum Publishers, New York, 1999), pp. 1–5. [7] B. van der Pol, Bull. Am. Math. Soc. 53, 976 (1947). [8] M. V. Berry, J. Phys. A: Math. Theor. 45, 302001 (2012). [9] C. Feiler and W. P. Schleich, New J. Phys. 15, 063009 (2013). [10] G. Polya, ´ Acta Math. 48, 305 (1926). [11] H. M. Edwards, Riemann’s Zeta Function (Academic Press, New York, 1974). [12] F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Phys. Rev. Lett. 67, 516 (1991). [13] M. Grifoni and P. Hanggi, ¨ Phys. Rep. 304, 229 (1998). [14] C. E. Creffield, Phys. Rev. B 67, 165301 (2003). [15] J. H. Shirley, Phys. Rev. 138, B979 (1965). [16] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403 (2007). [17] E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M. K. Oberthaler, Phys. Rev. Lett. 100, 190405 (2008). [18] A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. A 79, 013611 (2009). [19] Equation (4) contains a extra factor of four as compared with a similar expression appearing in Ref. [20] in order to agree with the original expression used by Pólya in Ref. [ 10]. [20] E. C. Titchmarsh, The Theory of the Riemann Zeta Function (Oxford University Press, Oxford, 1986). [21] G. Sierra and J. Rodr´ıguez-Laguna, Phys. Rev. Lett. 106, 200201 (2011). [22] G. Sierra, J. Phys. A: Math. Theor. 47, 325204 (2014). [23] Since we employ the relation F(t) = cos_(−1) φ(t), we clearly require |φ(t)|≤ 1 for all t. It can be shown that a necessary and sufficient condition for this is a^2 < x, where x and a parametrize the modified Bessel function (4). For the case we are interested in, a = 9/4 and x = 2π (e.g., 2.25 < 2.50), the relation a < √x is indeed satisfied. [24] C. E. Creffield, F. Sols, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. A 82, 035601 (2010). [25] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014). [26] A. M. Odlyzko, Numer. Algorithms 25, 293 (2000). | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 3b58cb19-3165-4b80-a65d-1e03b90ebf64 | |
| relation.isAuthorOfPublication.latestForDiscovery | 3b58cb19-3165-4b80-a65d-1e03b90ebf64 |
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