1Institute for Theoretical Physics, ETH Zürich, Switzerland
2Group of Applied Physics, University of Geneva, Switzerland
3Institute for Quantum Optics and Quantum Information (IQOQI), Vienna, Austria
4Department of Computer Science, University College London, United Kingdom
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
The theory of relativity associates a proper time with each moving object via its world line. In quantum theory however, such well-defined trajectories are forbidden. After introducing a general characterisation of quantum clocks, we demonstrate that, in the weak-field, low-velocity limit, all “good” quantum clocks experience time dilation as dictated by general relativity when their state of motion is classical (i.e. Gaussian). For nonclassical states of motion, on the other hand, we find that quantum interference effects may give rise to a significant discrepancy between the proper time and the time measured by the clock. The universality of this discrepancy implies that it is not simply a systematic error, but rather a quantum modification to the proper time itself. We also show how the clock’s delocalisation leads to a larger uncertainty in the time it measures – a consequence of the unavoidable entanglement between the clock time and its center-of-mass degrees of freedom. We demonstrate how this lost precision can be recovered by performing a measurement of the clock’s state of motion alongside its time reading.
Popular summary
determined by their path through spacetime — an effect known as time
dilation. Quantum mechanics, on the other hand, says that such definite
paths are impossible, and that all objects will instead follow
superpositions of different paths. What time, then, does a quantum clock
measure?
In this paper, we give an answer to this question. We use the simple fact
that every clock must have some internal workings whose changes track the
passage of time. This internal structure is associated with some energy,
and general relativity tells us how this energy interacts with the energy
of the clock’s motion, in turn determining the time experienced by the
clock. We can separate this effect into a part which depends on the
particulars of the clock, and a part which doesn’t. Since the latter is the
same for any system regardless of its makeup, it is a universal time
dilation.
For the kinds of motion that most resemble the “classical” paths used in
general relativity, we find that a clock measures (on average) the
classical time dilation. For a superposition of such paths, however, we
predict a classical and a quantum time dilation effect, and the quantum
part is large enough to potentially be observed in the near future. We also
find that this relationship between the energy of the internal working of
the clock and its motion generates quantum entanglement between them,
reducing the accuracy of measurements of the clock time (unless the clock’s
motion is measured too).
► BibTeX data
► References
[1] Isham, C. J. Canonical quantum gravity and the problem of time. In Integrable systems, quantum groups, and quantum field theories, 157–287 (Springer, 1993). URL https://doi.org/10.1007/978-94-011-1980-1_6.
https://doi.org/10.1007/978-94-011-1980-1_6
[2] Kuchař, K. V. Time and interpretations of quantum gravity. In 4th Canadian Conference on General Relativity and Relativistic Astrophysics, 211–314 (World Scientific, 1992). URL https://doi.org/10.1142/S0218271811019347.
https://doi.org/10.1142/S0218271811019347
[3] Anderson, E. Quantum Mechanics Versus General Relativity (Springer International Publishing, 2017), 1st edn. URL https://doi.org/10.1007/978-3-319-58848-3.
https://doi.org/10.1007/978-3-319-58848-3
[4] Lock, M. P. E. & Fuentes, I. Relativistic quantum clocks. In Time in Physics, 51–68 (Springer, 2017). URL https://doi.org/10.1007.
https://doi.org/10.1007
[5] Pauli, W. Die allgemeinen prinzipien der wellenmechanik. Handbuch der Physik 5, 1–168 (1958).
[6] Woods, M. P., Silva, R. & Oppenheim, J. Autonomous Quantum Machines and Finite-Sized Clocks. Annales Henri Poincaré (2018). URL https://doi.org/10.1007/s00023-018-0736-9.
https://doi.org/10.1007/s00023-018-0736-9
[7] Einstein, A. Zur elektrodynamik bewegter körper. Annalen der physik 322, 891–921 (1905). URL https://doi.org/10.1002/andp.19053221004.
https://doi.org/10.1002/andp.19053221004
[8] Bridgman, P. W. The logic of modern physics (Macmillan New York, 1927).
[9] Bužek, V., Derka, R. & Massar, S. Optimal quantum clocks. Phys. Rev. Lett. 82, 2207–2210 (1999). URL https://doi.org/10.1103/PhysRevLett.82.2207.
https://doi.org/10.1103/PhysRevLett.82.2207
[10] Erker, P. The Quantum Hourglass: approaching time measurement with quantum information theory. Master’s thesis, ETH Zürich (2014). URL https://doi.org/10.3929/ethz-a-010514644.
https://doi.org/10.3929/ethz-a-010514644
[11] Woods, M. P., Silva, R., Pütz, G., Stupar, S. & Renner, R. Quantum clocks are more accurate than classical ones. arXiv: 1806.00491 (2018). URL https://arxiv.org/abs/1806.00491.
arXiv:1806.00491
[12] Salecker, H. & Wigner, E. P. Quantum limitations of the measurement of space-time distances. Phys. Rev. 109, 571–577 (1958). URL https://doi.org/10.1007/978-3-662-09203-3_15.
https://doi.org/10.1007/978-3-662-09203-3_15
[13] Lindkvist, J. et al. Twin paradox with macroscopic clocks in superconducting circuits. Phys. Rev. A 90, 052113 (2014). URL https://doi.org/10.1103/PhysRevA.90.052113.
https://doi.org/10.1103/PhysRevA.90.052113
[14] Lorek, K., Louko, J. & Dragan, A. Ideal clocks — a convenient fiction. Class. Quantum Gravity 32, 175003 (2015). URL https://doi.org/10.1088/0264-9381/32/17/175003.
https://doi.org/10.1088/0264-9381/32/17/175003
[15] Lock, M. P. E. & Fuentes, I. Quantum and classical effects in light-clock falling in schwarzschild geometry. Class. Quantum Gravity 36, 175007 (2019). URL https://doi.org/10.1088/1361-6382/ab32b1.
https://doi.org/10.1088/1361-6382/ab32b1
[16] Anastopoulos, C. & Hu, B.-L. Equivalence principle for quantum systems: dephasing and phase shift of free-falling particles. Class. Quantum Gravity 35, 035011 (2018). URL https://doi.org/10.1088/1361-6382/aaa0e8.
https://doi.org/10.1088/1361-6382/aaa0e8
[17] Zych, M. Quantum Systems under Gravitational Time Dilation. Springer Theses (Springer, 2017). URL https://doi.org/10.1007/978-3-319-53192-2.
https://doi.org/10.1007/978-3-319-53192-2
[18] Lämmerzahl, C. A Hamilton operator for quantum optics in gravitational fields. Phys. Lett. A 203, 12–17 (1995). URL https://doi.org/10.1016/0375-9601(95)00345-4.
https://doi.org/10.1016/0375-9601(95)00345-4
[19] Zych, M., Costa, F., Pikovski, I. & Brukner, Č. Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun. 2, 505 (2011). URL https://doi.org/10.1038/ncomms1498.
https://doi.org/10.1038/ncomms1498
[20] Pikovski, I., Zych, M., Costa, F. & Brukner, Č. Universal decoherence due to gravitational time dilation. Nat. Phys. 11, 668–672 (2015). URL https://doi.org/10.1038/nphys3366.
https://doi.org/10.1038/nphys3366
[21] Garrison, J. C. & Wong, J. Canonically conjugate pairs, uncertainty relations, and phase operators. J. Math. Phys. 11, 2242–2249 (1970). URL https://doi.org/10.1063/1.1665388.
https://doi.org/10.1063/1.1665388
[22] Srinivas, M. & Vijayalakshmi, R. The ‘time of occurrence’ in quantum mechanics. Pramana 16, 173–199 (1981). URL https://doi.org/10.1007/BF02848181.
https://doi.org/10.1007/BF02848181
[23] Holevo, A. Covariant measurements and uncertainty relations. Reports on Mathematical Physics 16, 385 – 400 (1979). URL https://doi.org/10.1016/0034-4877(79)90072-7.
https://doi.org/10.1016/0034-4877(79)90072-7
[24] Peres, A. Measurement of time by quantum clocks. Am. J. Phys 48, 552 (1980). URL https://doi.org/10.1119/1.12061.
https://doi.org/10.1119/1.12061
[25] Weinberg, S. Gravitation and cosmology: Principles and Applications of the General Theory of Relativity. ed. John Wiley and Sons, New York (1972). URL https://www.wiley.com/en-us/Gravitation+and+Cosmology.
https://www.wiley.com/en-us/Gravitation+and+Cosmology
[26] Zych, M. & Brukner, Č. Quantum formulation of the einstein equivalence principle. Nat. Phys. 1 (2018). URL https://doi.org/10.1038/s41567-018-0197-6.
https://doi.org/10.1038/s41567-018-0197-6
[27] Hudson, R. L. When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249–252 (1974). URL https://doi.org/10.1016/0034-4877(74)90007-X.
https://doi.org/10.1016/0034-4877(74)90007-X
[28] Woods, M. P. & Alhambra, Á. M. Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames. Quantum 4, 245 (2020). URL https://doi.org/10.22331/q-2020-03-23-245.
https://doi.org/10.22331/q-2020-03-23-245
[29] Chou, C.-W., Hume, D., Rosenband, T. & Wineland, D. Optical clocks and relativity. Science 329, 1630–1633 (2010). URL https://doi.org/10.1126/science.1192720.
https://doi.org/10.1126/science.1192720
[30] Brewer, S. M. et al. $^{27}{mathrm{al}}^{+}$ quantum-logic clock with a systematic uncertainty below ${10}^{{-}18}$. Phys. Rev. Lett. 123, 033201 (2019). URL https://doi.org/10.1103/PhysRevLett.123.033201.
https://doi.org/10.1103/PhysRevLett.123.033201
[31] Mantina, M., Chamberlin, A. C., Valero, R., Cramer, C. J. & Truhlar, D. G. Consistent van der waals radii for the whole main group. J. Phys. Chem. A 113, 5806–5812 (2009). URL https://doi.org/10.1021/jp8111556.
https://doi.org/10.1021/jp8111556
[32] Ludlow, A. D., Boyd, M. M., Ye, J., Peik, E. & Schmidt, P. O. Optical atomic clocks. Rev. Mod. Phys. 87, 637 (2015). URL https://doi.org/10.1103/RevModPhys.87.637.
https://doi.org/10.1103/RevModPhys.87.637
[33] Kovachy, T. et al. Quantum superposition at the half-metre scale. Nature 528, 530 (2015). URL https://doi.org/10.1038/nature16155.
https://doi.org/10.1038/nature16155
[34] Page, D. N. & Wootters, W. K. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D 27, 2885 (1983). URL https://doi.org/10.1103/PhysRevD.27.2885.
https://doi.org/10.1103/PhysRevD.27.2885
[35] Smith, A. R. H. & Ahmadi, M. Relativistic quantum clocks observe classical and quantum time dilation. arXiv:1904.12390 (2019). URL https://arxiv.org/abs/1904.12390.
arXiv:1904.12390
[36] Hoehn, P. A., Smith, A. R. H. & Lock, M. P. E. Equivalence of approaches to relational quantum dynamics in relativistic settings. arXiv:2007.00580 (2020). URL https://arxiv.org/abs/2007.00580.
arXiv:2007.00580
[37] Colella, R., Overhauser, A. W. & Werner, S. A. Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472 (1975). URL https://doi.org/10.1103/PhysRevLett.34.1472.
https://doi.org/10.1103/PhysRevLett.34.1472
[38] Roura, A. Gravitational redshift in quantum-clock interferometry. Phys. Rev. X 10, 021014 (2020). URL https://doi.org/10.1103/PhysRevX.10.021014.
https://doi.org/10.1103/PhysRevX.10.021014
[39] Peters, Achim and Chung, Keng Yeow and Chu, Steven. Measurement of gravitational acceleration by dropping atoms. Nature 400, 849–852 (1999). URL https://doi.org/10.1038/23655.
https://doi.org/10.1038/23655
[40] Müller, H., Peters, A. & Chu, S. A precision measurement of the gravitational redshift by the interference of matter waves. Nature 463, 926–929 (2010). URL https://doi.org/10.1038/nature08776.
https://doi.org/10.1038/nature08776
[41] Wolf, P. et al. Atom gravimeters and gravitational redshift. Nature 467, E1–E1 (2010). URL https://doi.org/10.1038/nature09340.
https://doi.org/10.1038/nature09340
[42] Does an atom interferometer test the gravitational redshift at the compton frequency? URL https://doi.org/10.1088.
https://doi.org/10.1088
[43] Sinha, S. & Samuel, J. Atom interferometry and the gravitational redshift. Class. Quantum Gravity 28, 145018 (2011). URL https://doi.org/10.1088.
https://doi.org/10.1088
[44] Ranković, S., Liang, Y.-C. & Renner, R. Quantum clocks and their synchronisation – the Alternate Ticks Game. arXiv:1506.01373 (2015). URL http://arxiv.org/abs/1506.01373.
arXiv:1506.01373
[45] Erker, P. et al. Autonomous quantum clocks: does thermodynamics limit our ability to measure time? Phys. Rev. X 7, 031022 (2017). URL https://doi.org/10.1103/PhysRevX.7.031022.
https://doi.org/10.1103/PhysRevX.7.031022
[46] Woods, M. P. Autonomous Ticking Clocks from Axiomatic Principles. arXiv: 2005.04628 (2020). URL https://arxiv.org/abs/2005.04628.
arXiv:2005.04628
[47] Schwarzhans, E., Lock, M. P. E., Erker, P., Friis, N. & Huber, M. Autonomous temporal probability concentration: Clockworks and the second law of thermodynamics. arXiv:2007.01307 (2020). URL https://arxiv.org/abs/2007.01307.
arXiv:2007.01307
[48] Życzkowski, K., Horodecki, P., Sanpera, A. & Lewenstein, M. Volume of the set of separable states. Phys. Rev. A 58, 883 (1998). URL https://doi.org/10.1103/PhysRevA.58.883.
https://doi.org/10.1103/PhysRevA.58.883
[49] Castro-Ruiz, E., Giacomini, F. & Brukner, Č. Entanglement of quantum clocks through gravity. Proc. Natl. Acad. Sci. U.S.A. 201616427 (2017). URL https://doi.org/10.1073/pnas.1616427114.
https://doi.org/10.1073/pnas.1616427114
[50] Paige, A. J., Plato, A. D. K. & Kim, M. S. Classical and nonclassical time dilation for quantum clocks. Phys. Rev. Lett. 124, 160602 (2020). URL https://doi.org/10.1103/PhysRevLett.124.160602.
https://doi.org/10.1103/PhysRevLett.124.160602
[51] Pikovski, I., Zych, M., Costa, F. & Brukner, C. Time Dilation in Quantum Systems and Decoherence: Questions and Answers. arXiv:1508.03296 (2015). URL http://arxiv.org/abs/1508.03296.
arXiv:1508.03296
[52] Shankar, R. Principles of Quantum Mechanics (Springer US, Boston, MA, 1994). URL https://doi.org/10.1007/978-1-4757-0576-8.
https://doi.org/10.1007/978-1-4757-0576-8
[53] Hall, B. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics (Springer International Publishing, 2015). URL https://doi.org/10.1007/978-3-319-13467-3.
https://doi.org/10.1007/978-3-319-13467-3
[54] Hoehn, P. A., Smith, A. R. H. & Lock, M. P. E. The Trinity of Relational Quantum Dynamics. arXiv:1912.00033 (2019). URL https://arxiv.org/abs/1912.00033.
arXiv:1912.00033
[55] Holevo, A. S. Probabilistic and Statistical Aspects of Quantum Theory, vol. 1 (North-Holland, Amsterdam, 1982). URL https://doi.org/10.1007/978-88-7642-378-9.
https://doi.org/10.1007/978-88-7642-378-9
[56] Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nature photonics 5, 222 (2011). URL https://doi.org/10.1038/nphoton.2011.35.
https://doi.org/10.1038/nphoton.2011.35
Cited by
[1] Alexander R. H. Smith and Mehdi Ahmadi, “Relativistic quantum clocks observe classical and quantum time dilation”, arXiv:1904.12390.
[2] Maximilian P. E. Lock and Ivette Fuentes, “Quantum and classical effects in a light-clock falling in Schwarzschild geometry”, Classical and Quantum Gravity 36 17, 175007 (2019).
[3] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, “The Trinity of Relational Quantum Dynamics”, arXiv:1912.00033.
[4] Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner, “Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems”, Nature Communications 11, 2672 (2020).
[5] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, “Equivalence of approaches to relational quantum dynamics in relativistic settings”, arXiv:2007.00580.
[6] Alexander R. H. Smith, “Quantum time dilation: A new test of relativistic quantum theory”, arXiv:2004.10810.
[7] Mischa P. Woods, “Autonomous Ticking Clocks from Axiomatic Principles”, arXiv:2005.04628.
[8] Yuxiang Yang and Renato Renner, “Ultimate limit on time signal generation”, arXiv:2004.07857.
[9] Piotr T. Grochowski, Alexander R. H. Smith, Andrzej Dragan, and Kacper Dębski, “Quantum time dilation in atomic spectra”, arXiv:2006.10084.
[10] Carolyn E. Wood and Magdalena Zych, “Minimum uncertainty states for free particles with quantized mass-energy”, arXiv:1911.06653.
The above citations are from SAO/NASA ADS (last updated successfully 2020-08-17 03:49:56). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2020-08-17 03:49:55).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
Source: https://quantum-journal.org/papers/q-2020-08-14-309/
