{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T00:54:43Z","timestamp":1773104083675,"version":"3.50.1"},"reference-count":57,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,5,7]],"date-time":"2024-05-07T00:00:00Z","timestamp":1715040000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"(Polish) National Science Center","award":["2020\/36\/T\/ST2\/00065"],"award-info":[{"award-number":["2020\/36\/T\/ST2\/00065"]}]},{"name":"(Polish) National Science Center","award":["2021\/41\/N\/ST2\/01901"],"award-info":[{"award-number":["2021\/41\/N\/ST2\/01901"]}]},{"DOI":"10.13039\/501100001870","name":"Foundation for Polish Science (FNP).","doi-asserted-by":"crossref","award":["START"],"award-info":[{"award-number":["START"]}],"id":[{"id":"10.13039\/501100001870","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Saint Anselm College","award":["summer research grant"],"award-info":[{"award-number":["summer research grant"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"According to relativity, the reading of an ideal clock is interpreted as the elapsed proper time along its classical trajectory through spacetime. In contrast, quantum theory allows the association of many simultaneous trajectories with a single quantum clock, each weighted appropriately. Here, we investigate how the superposition principle affects the gravitational time dilation observed by a simple clock \u2013 a decaying two-level atom. Placing such an atom in a superposition of positions enables us to analyze a quantum contribution to a classical time dilation manifest in spontaneous emission. In particular, we show that the emission rate of an atom prepared in a coherent superposition of separated wave packets in a gravitational field is different from the emission rate of an atom in a classical mixture of these packets, which gives rise to a quantum gravitational time dilation effect. We demonstrate that this nonclassical effect also manifests in a fractional frequency shift of the internal energy of the atom that is within the resolution of current atomic clocks. In addition, we show the effect of spatial coherence on the atom&apos;s emission spectrum.<\/jats:p>","DOI":"10.22331\/q-2024-05-07-1338","type":"journal-article","created":{"date-parts":[[2024,5,7]],"date-time":"2024-05-07T11:19:21Z","timestamp":1715080761000},"page":"1338","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Quantum time dilation in a gravitational field"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0674-9819","authenticated-orcid":false,"given":"Jerzy","family":"Paczos","sequence":"first","affiliation":[{"name":"Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8865-9066","authenticated-orcid":false,"given":"Kacper","family":"D\u0119bski","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9654-4824","authenticated-orcid":false,"given":"Piotr T.","family":"Grochowski","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\u00f3w 32\/46, 02-668 Warsaw, Poland"},{"name":"Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4618-4832","authenticated-orcid":false,"given":"Alexander R. H.","family":"Smith","sequence":"additional","affiliation":[{"name":"Department of Physics, Saint Anselm College, Manchester, New Hampshire 03102, USA"},{"name":"Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5254-710X","authenticated-orcid":false,"given":"Andrzej","family":"Dragan","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"},{"name":"Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore"}]}],"member":"9598","published-online":{"date-parts":[[2024,5,7]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. Colella, A. W. Overhauser, and S. A. Werner. ``Observation of gravitationally induced quantum interference''. Phys. Rev. Lett. 34, 1472 (1975).","DOI":"10.1103\/PhysRevLett.34.1472"},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. Peters, K. Chung, and S. Chu. ``Measurement of gravitational acceleration by dropping atoms''. Nature 400, 849 (1999).","DOI":"10.1038\/23655"},{"key":"2","doi-asserted-by":"publisher","unstructured":"H. M\u00fcller, A. Peters, and S. Chu. ``A precision measurement of the gravitational redshift by the interference of matter waves''. Nature 463, 926 (2010).","DOI":"10.1038\/nature08776"},{"key":"3","doi-asserted-by":"publisher","unstructured":"V. Vedral and F. Morikoshi. ``Schr\u00f6dinger's Cat Meets Einstein's Twins: A Superposition of Different Clock Times''. Int. J. Theor. Phys. 47, 2126 (2008).","DOI":"10.1007\/s10773-007-9568-y"},{"key":"4","doi-asserted-by":"publisher","unstructured":"M. Zych, F. Costa, I. Pikovski, and \u010c. Brukner. ``Quantum interferometric visibility as a witness of general relativistic proper time''. Nat. Commun. 2, 505 (2011).","DOI":"10.1038\/ncomms1498"},{"key":"5","doi-asserted-by":"publisher","unstructured":"P. A. Bushev, J. H. Cole, D. Sholokhov, N. Kukharchyk, and M. Zych. ``Single electron relativistic clock interferometer''. New J. Phys. 18, 093050 (2016).","DOI":"10.1088\/1367-2630\/18\/9\/093050"},{"key":"6","doi-asserted-by":"publisher","unstructured":"S. Loriani et al. ``Interference of clocks: A quantum twin paradox''. Sci. Adv. 5, aax8966 (2019).","DOI":"10.1126\/sciadv.aax8966"},{"key":"7","doi-asserted-by":"publisher","unstructured":"A. Roura. ``Gravitational Redshift in Quantum-Clock Interferometry''. Phys. Rev. X 10, 021014 (2020).","DOI":"10.1103\/PhysRevX.10.021014"},{"key":"8","doi-asserted-by":"publisher","unstructured":"J. Lindkvist, C. Sab\u00edn, I. Fuentes, A. Dragan, I. Svensson, P. Delsing, and G. Johansson. ``Twin paradox with macroscopic clocks in superconducting circuits''. Phys. Rev. A 90, 052113 (2014).","DOI":"10.1103\/PhysRevA.90.052113"},{"key":"9","doi-asserted-by":"publisher","unstructured":"K. Lorek, J. Louko, and A. Dragan. ``Ideal clocks\u2014 a convenient fiction''. Class. Quantum Grav. 32, 175003 (2015).","DOI":"10.1088\/0264-9381\/32\/17\/175003"},{"key":"10","doi-asserted-by":"publisher","unstructured":"A. J. Paige, A. D. K. Plato, and M. S. Kim. ``Classical and Nonclassical Time Dilation for Quantum Clocks''. Phys. Rev. Lett. 124, 160602 (2020).","DOI":"10.1103\/PhysRevLett.124.160602"},{"key":"11","doi-asserted-by":"publisher","unstructured":"A. R. H. Smith and M. Ahmadi. ``Quantum clocks observe classical and quantum time dilation''. Nat. Commun. 11, 5360 (2020).","DOI":"10.1038\/s41467-020-18264-4"},{"key":"12","doi-asserted-by":"publisher","unstructured":"A. Holevo. ``Probabilistic and Statistical Aspects of Quantum Theory''. Edizioni della Normale. Pisa (2011).","DOI":"10.1007\/978-88-7642-378-9"},{"key":"13","doi-asserted-by":"publisher","unstructured":"P. Busch, M. Grabowski, and P. J. Lahti. ``Operational Quantum Physics''. Springer. Berlin, Heidelberg (1995).","DOI":"10.1007\/978-3-540-49239-9"},{"key":"14","doi-asserted-by":"publisher","unstructured":"F. Giacomini and A. Kempf. ``Second-quantized Unruh-DeWitt detectors and their quantum reference frame transformations''. Phys. Rev. D 105, 125001 (2022).","DOI":"10.1103\/PhysRevD.105.125001"},{"key":"15","doi-asserted-by":"publisher","unstructured":"P. T. Grochowski, A. R. H. Smith, A. Dragan, and K. D\u0119bski. ``Quantum time dilation in atomic spectra''. Phys. Rev. Res. 3, 023053 (2021).","DOI":"10.1103\/PhysRevResearch.3.023053"},{"key":"16","doi-asserted-by":"publisher","unstructured":"S. Khandelwal, M. P. E. Lock, and M. P. Woods. ``Universal quantum modifications to general relativistic time dilation in delocalised clocks''. Quantum 4, 309 (2020).","DOI":"10.22331\/q-2020-08-14-309"},{"key":"17","doi-asserted-by":"publisher","unstructured":"F. Giacomini. ``Spacetime Quantum Reference Frames and superpositions of proper times''. Quantum 5, 508 (2021).","DOI":"10.22331\/q-2021-07-22-508"},{"key":"18","doi-asserted-by":"publisher","unstructured":"N. Stritzelberger and A. Kempf. ``Coherent delocalization in the light-matter interaction''. Phys. Rev. D 101, 036007 (2020).","DOI":"10.1103\/PhysRevD.101.036007"},{"key":"19","doi-asserted-by":"publisher","unstructured":"R. Lopp and E. Mart\u00edn-Mart\u00ednez. ``Quantum delocalization, gauge, and quantum optics: Light-matter interaction in relativistic quantum information''. Phys. Rev. A 103, 013703 (2021).","DOI":"10.1103\/PhysRevA.103.013703"},{"key":"20","doi-asserted-by":"publisher","unstructured":"K. Rz\u0105\u017cewski and W. \u017bakowicz. ``Spontaneous emission from an extended wavepacket''. J. Phys. B 25, L319 (1992).","DOI":"10.1088\/0953-4075\/25\/13\/001"},{"key":"21","doi-asserted-by":"publisher","unstructured":"B. Maybee, D. Hodgson, A. Beige, and R. Purdy. ``A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes''. Entropy 21, 844 (2019).","DOI":"10.3390\/e21090844"},{"key":"22","doi-asserted-by":"publisher","unstructured":"M. Kasevich and S. Chu. ``Measurement of the gravitational acceleration of an atom with a light-pulse atom interferometer''. Appl. Phys. B 54, 321 (1992).","DOI":"10.1007\/BF00325375"},{"key":"23","doi-asserted-by":"publisher","unstructured":"I. Pikovski, M. Zych, F. Costa, and \u010c. Brukner. ``Universal decoherence due to gravitational time dilation''. Nat. Phys. 11, 668 (2015).","DOI":"10.1038\/nphys3366"},{"key":"24","doi-asserted-by":"publisher","unstructured":"B. H. Pang, Y. Chen, and F. Y. Khalili. ``Universal Decoherence under Gravity: A Perspective through the Equivalence Principle''. Phys. Rev. Lett. 117, 090401 (2016).","DOI":"10.1103\/PhysRevLett.117.090401"},{"key":"25","doi-asserted-by":"publisher","unstructured":"D. Gao, J. Wang, and M. Zhan. ``Atomic Interferometric Gravitational-Wave Space Observatory (AIGSO)''. Commun. Theor. Phys. 69, 37 (2018).","DOI":"10.1088\/0253-6102\/69\/1\/37"},{"key":"26","doi-asserted-by":"publisher","unstructured":"D. Schlippert et al. ``Quantum Test of the Universality of Free Fall''. Phys. Rev. Lett. 112, 203002 (2014).","DOI":"10.1103\/PhysRevLett.112.203002"},{"key":"27","doi-asserted-by":"publisher","unstructured":"L. Viola and R. Onofrio. ``Testing the equivalence principle through freely falling quantum objects''. Phys. Rev. D 55, 455\u2013462 (1997).","DOI":"10.1103\/PhysRevD.55.455"},{"key":"28","doi-asserted-by":"publisher","unstructured":"C. Anastopoulos and B. L. Hu. ``Equivalence principle for quantum systems: Dephasing and phase shift of free-falling particles''. Class. Quantum Grav. 35, 035011 (2018).","DOI":"10.1088\/1361-6382\/aaa0e8"},{"key":"29","doi-asserted-by":"publisher","unstructured":"M. Zych and \u010c. Brukner. ``Quantum formulation of the Einstein equivalence principle''. Nat. Phys. 14, 1027\u20131031 (2018).","DOI":"10.1038\/s41567-018-0197-6"},{"key":"30","doi-asserted-by":"publisher","unstructured":"M. Sonnleitner, N. Trautmann, and S. M. Barnett. ``Will a Decaying Atom Feel a Friction Force?''. Phys. Rev. Lett. 118, 053601 (2017).","DOI":"10.1103\/PhysRevLett.118.053601"},{"key":"31","doi-asserted-by":"publisher","unstructured":"M. Sonnleitner and S. M. Barnett. ``Mass-energy and anomalous friction in quantum optics''. Phys. Rev. A 98, 042106 (2018).","DOI":"10.1103\/PhysRevA.98.042106"},{"key":"32","doi-asserted-by":"publisher","unstructured":"K. Marzlin. ``Dipole coupling of atoms and light in gravitational fields''. Phys. Rev. A 51, 625\u2013631 (1995).","DOI":"10.1103\/PhysRevA.51.625"},{"key":"33","doi-asserted-by":"publisher","unstructured":"P. K. Schwartz and D. Giulini. ``Post-Newtonian Hamiltonian description of an atom in a weak gravitational field''. Phys. Rev. A 100, 052116 (2019).","DOI":"10.1103\/PhysRevA.100.052116"},{"key":"34","doi-asserted-by":"publisher","unstructured":"V. J. Mart\u00ednez-Lahuerta, S. Eilers, T. E. Mehlst\u00e4ubler, P. O. Schmidt, and K. Hammerer. ``Ab initio quantum theory of mass defect and time dilation in trapped-ion optical clocks''. Phys. Rev. A 106, 032803 (2022).","DOI":"10.1103\/PhysRevA.106.032803"},{"key":"35","doi-asserted-by":"publisher","unstructured":"M. Zych, \u0141. Rudnicki, and I. Pikovski. ``Gravitational mass of composite systems''. Phys. Rev. D 99, 104029 (2019).","DOI":"10.1103\/PhysRevD.99.104029"},{"key":"36","doi-asserted-by":"publisher","unstructured":"C. L\u00e4mmerzahl. ``A Hamilton operator for quantum optics in gravitational fields''. Phys. Lett. A 203, 12\u201317 (1995).","DOI":"10.1016\/0375-9601(95)00345-4"},{"key":"37","doi-asserted-by":"publisher","unstructured":"A. Dragan. ``Unusually Special Relativity''. World Scientific. (2021).","DOI":"10.1142\/q0319"},{"key":"38","doi-asserted-by":"publisher","unstructured":"C. Gerry and P. Knight. ``Introductory Quantum Optics''. Cambridge University Press. Cambridge (2004).","DOI":"10.1017\/CBO9780511791239"},{"key":"39","doi-asserted-by":"publisher","unstructured":"P. Lambropoulos and D. Petrosyan. ``Fundamentals of Quantum Optics and Quantum Information''. Springer. Berlin, Heidelberg (2007).","DOI":"10.1007\/978-3-540-34572-5"},{"key":"40","doi-asserted-by":"publisher","unstructured":"G. Grynberg, A. Aspect, and C. Fabre. ``Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light''. Cambridge University Press. Cambridge (2010).","DOI":"10.1017\/CBO9780511778261"},{"key":"41","doi-asserted-by":"publisher","unstructured":"C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland. ``Optical Clocks and Relativity''. Science 329, 1630\u20131633 (2010).","DOI":"10.1126\/science.1192720"},{"key":"42","doi-asserted-by":"publisher","unstructured":"J. C. Hafele and Richard E. Keating. ``Around-the-World Atomic Clocks: Predicted Relativistic Time Gains''. Science 177, 166\u2013168 (1972).","DOI":"10.1126\/science.177.4044.166"},{"key":"43","doi-asserted-by":"publisher","unstructured":"M. Takamoto et al. ``Test of general relativity by a pair of transportable optical lattice clocks''. Nat. Photonics 14, 411\u2013415 (2020).","DOI":"10.1038\/s41566-020-0619-8"},{"key":"44","doi-asserted-by":"publisher","unstructured":"P. Laurent, D. Massonnet, L. Cacciapuoti, and C. Salomon. ``The ACES\/PHARAO space mission''. C. R. Phys. 16, 540\u2013552 (2015).","DOI":"10.1016\/j.crhy.2015.05.002"},{"key":"45","doi-asserted-by":"publisher","unstructured":"G. M. Tino et al. ``SAGE: A proposal for a space atomic gravity explorer''. Eur. Phys. J. D 73, 228 (2019).","DOI":"10.1140\/epjd\/e2019-100324-6"},{"key":"46","doi-asserted-by":"publisher","unstructured":"T. Bothwell et al. ``Resolving the gravitational redshift across a millimetre-scale atomic sample''. Nature 602, 420\u2013424 (2022).","DOI":"10.1038\/s41586-021-04349-7"},{"key":"47","doi-asserted-by":"publisher","unstructured":"X. Zheng, J. Dolde, V. Lochab, B. N. Merriman, H. Li, and S. Kolkowitz. ``Differential clock comparisons with a multiplexed optical lattice clock''. Nature 602, 425\u2013430 (2022).","DOI":"10.1038\/s41586-021-04344-y"},{"key":"48","doi-asserted-by":"publisher","unstructured":"R. Charri\u00e8re, M. Cadoret, N. Zahzam, Y. Bidel, and A. Bresson. ``Local gravity measurement with the combination of atom interferometry and bloch oscillations''. Phys. Rev. A 85, 013639 (2012).","DOI":"10.1103\/PhysRevA.85.013639"},{"key":"49","doi-asserted-by":"publisher","unstructured":"X. Zhang, R. P. del Aguila, T. Mazzoni, N. Poli, and G. M. Tino. ``Trapped-atom interferometer with ultracold sr atoms''. Phys. Rev. A 94, 043608 (2016).","DOI":"10.1103\/PhysRevA.94.043608"},{"key":"50","doi-asserted-by":"publisher","unstructured":"V. Xu, M. Jaffe, C. D. Panda, S. L. Kristensen, L. W. Clark, and H. M\u00fcller. ``Probing gravity by holding atoms for 20 seconds''. Science 366, 745\u2013749 (2019).","DOI":"10.1126\/science.aay6428"},{"key":"51","unstructured":"C. D. Panda, M. Tao, J. Egelhoff, M. Ceja, V. Xu, and H. M\u00fcller. ``Minute-scale gravimetry using a coherent atomic spatial superposition'' (2023). arXiv:2210.07289."},{"key":"52","doi-asserted-by":"publisher","unstructured":"R. J. Rafac, C. E. Tanner, A. E. Livingston, K. W. Kukla, H. G. Berry, and C. A. Kurtz. ``Precision lifetime measurements of the $6_{p^2 p_{1\/2,3\/2}}$ states in atomic cesium''. Phys. Rev. A 50, R1976(R) (1994).","DOI":"10.1103\/PhysRevA.50.R1976"},{"key":"53","doi-asserted-by":"publisher","unstructured":"D. L. Moehring et al. ``Precision lifetime measurements of a single trapped ion with ultrafast laser pulses''. Phys. Rev. A 73, 023413 (2006).","DOI":"10.1103\/PhysRevA.73.023413"},{"key":"54","doi-asserted-by":"publisher","unstructured":"M. Seidlitz et al. ``Precision lifetime measurements of the first 2${}^{+}$ and 4${}^{+}$ states in ${}^{56}$cr at the $n=32$ subshell closure''. Phys. Rev. C 84, 034318 (2011).","DOI":"10.1103\/PhysRevC.84.034318"},{"key":"55","doi-asserted-by":"publisher","unstructured":"R. B. Hutson, A. Goban, G. E. Marti, L. Sonderhouse, C. Sanner, and J. Ye. ``Engineering quantum states of matter for atomic clocks in shallow optical lattices''. Phys. Rev. Lett. 123, 123401 (2019).","DOI":"10.1103\/PhysRevLett.123.123401"},{"key":"56","doi-asserted-by":"publisher","unstructured":"X. Zheng, J. Dolde, M. C. Cambria, H. M. Lim, and S. Kolkowitz. ``A lab-based test of the gravitational redshift with a miniature clock network''. Nat. Commun. 14, 4886 (2023).","DOI":"10.1038\/s41467-023-40629-8"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-05-07-1338\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,5,7]],"date-time":"2024-05-07T11:19:32Z","timestamp":1715080772000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-05-07-1338\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,7]]},"references-count":57,"URL":"https:\/\/doi.org\/10.22331\/q-2024-05-07-1338","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,7]]},"article-number":"1338"}}