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Quantum time dilation in a gravitational field


Jerzy Paczos1, Kacper Dębski2, Piotr T. Grochowski3,4,5, Alexander R. H. Smith6,7, and Andrzej Dragan2,8

1Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden
2Institute of Theoretical Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
3Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
4Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
5Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
6Department of Physics, Saint Anselm College, Manchester, New Hampshire 03102, USA
7Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
8Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore

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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 – 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’s emission spectrum.

A hallmark prediction of Einstein’s general relativity is gravitational time dilation: clocks at different heights in a gravitational field tick at different rates. Quantum mechanics allows for the possibility of matter being in a superposition of two different locations, so it is natural to ask what time dilation a clock would experience if it were in a superposition of two heights in a gravitational field. We examine this question and show that the lifetime of an excited atom depends on whether it is in a quantum superposition or a classical mixture of heights in a gravitational field, leading to the phenomenon of quantum time dilation. In particular, we show that the fractional frequency shift of an atom’s spectrum, which is the observable measured by atomic clocks, is sensitive to this quantum time dilation effect. We estimate the resulting fractional frequency shift, concluding that its observation is within the precision of state-of-the-art atomic clocks and thus has the potential to offer a new test of fundamental physics at the intersection of quantum theory and gravitational physics.

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Cited by

[1] Kacper Dębski, Piotr T. Grochowski, Rafał Demkowicz-Dobrzański, and Andrzej Dragan, “Universality of quantum time dilation”, arXiv:2211.02425, (2022).

[2] Takeshi Chiba and Shunichiro Kinoshita, “Quantum clocks, gravitational time dilation, and quantum interference”, Physical Review D 106 12, 124035 (2022).

[3] Simone Rijavec, “Robustness of the Page-Wootters construction across different pictures, states of the universe, and system-clock interactions”, Physical Review D 108 6, 063507 (2023).

[4] Joshua Foo and Magdalena Zych, “Superpositions of thermalisation states in relativistic quantum field theory”, arXiv:2307.02593, (2023).

[5] Jerzy Paczos and Luis C. Barbado, “Hawking radiation for detectors in superposition of locations outside a black hole”, Physical Review D 108 12, 125015 (2023).

[6] Jiatong Yan and Baocheng Zhang, “Influence of gravitational waves on quantum multibody states”, Physical Review D 108 10, 105015 (2023).

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