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Testing the equivalence principle with time-diffracted free-falling quantum particles

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Abstract

The equivalence principle of gravity is examined at the quantum level using the diffraction in time of matter waves in two ways. First, we consider a quasi-monochromatic beam of particles incident on a shutter which is removed at time \(t = 0\) and fall due to the gravitational field. The probability density exhibits a set of mass-dependent oscillations which are genuinely quantum in nature, thereby reflecting quantum violations to the weak equivalence principle, although the strong equivalence principle remains valid. We estimate the degree of violation in terms of the width of the diffraction-in-time effect. Second, motivated by the recent advances in the manipulation of ultracold atoms and neutrons as well as the experimental observation of quantum states of ultracold neutrons in the gravitational field above a flat mirror, we study the diffraction in time of a suddenly released beam of particles initially prepared in gravitational quantum bound states. In this case, we quantify the degree of violation by comparing the time of flight from the mean position of the initial wave packet versus the time of flight as measured from the mirror. We show that, in this case both the weak and strong versions of the equivalence principle are violated. We demonstrate that compatibility between equivalence principle and quantum mechanics is recovered in the macroscopic (large-mass) limit. Possible realizations with ultracold neutrons, cesium atoms and large molecules are discussed.

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No datasets were generated or analyzed during the current study.

References

  1. R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)

    Book  Google Scholar 

  2. C.M. Will, The confrontation between general relativity and experiment. Living Rev. Relativ. 4, 4 (2001). https://doi.org/10.12942/lrr-2001-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. R. Colella, A.W. Overhauser, S.A. Werner, Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472–1474 (1975). https://doi.org/10.1103/PhysRevLett.34.1472

    Article  ADS  Google Scholar 

  4. A. Peters, K.Y. Chung, S. Chu, Measurement of gravitational acceleration by dropping atoms. Nature 400, 849–852 (1999). https://doi.org/10.1038/23655

    Article  ADS  Google Scholar 

  5. L. Viola, R. Onofrio, Testing the equivalence principle through freely falling quantum objects. Phys. Rev. D 55, 455–462 (1997). https://doi.org/10.1103/PhysRevD.55.455

    Article  ADS  Google Scholar 

  6. M.M. Ali, A.S. Majumdar, D. Home, A.K. Pan, On the quantum analogue of Galileo’s leaning tower experiment. Class. Quantum Gravity 23, 6493–6502 (2006). https://doi.org/10.1088/0264-9381/23/22/024

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. P. Chowdhury, D. Home, A.S. Majumdar, S.V. Mousavi, M.R. Mozaffari, S. Sinha, Strong quantum violation of the gravitational weak equivalence principle by a non-Gaussian wave packet. Class. Quantum Gravity 29, 025010 (2011). https://doi.org/10.1088/0264-9381/29/2/025010

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. R.S. Dumont, T.L. Marchioro II., Tunneling-time probability distribution. Phys. Rev. A 47, 85–97 (1993). https://doi.org/10.1103/PhysRevA.47.85

    Article  ADS  Google Scholar 

  9. W.R. McKinnon, C.R. Leavens, Distributions of delay times and transmission times in Bohm’s causal interpretation of quantum mechanics. Phys. Rev. A 51, 2748–2757 (1995). https://doi.org/10.1103/PhysRevA.51.2748

    Article  ADS  Google Scholar 

  10. V. Delgado, Quantum probability distribution of arrival times and probability current density. Phys. Rev. A 59, 1010–1020 (1999). https://doi.org/10.1103/PhysRevA.59.1010

    Article  ADS  Google Scholar 

  11. C.R. Leavens, Time of arrival in quantum and Bohmian mechanics. Phys. Rev. A 58, 840–847 (1998). https://doi.org/10.1103/PhysRevA.58.840

    Article  ADS  MathSciNet  Google Scholar 

  12. C.R. Leavens, Arrival time distributions. Phys. Lett. A 178, 27–32 (1993). https://doi.org/10.1016/0375-9601(93)90722-C

    Article  ADS  MATH  Google Scholar 

  13. J.G. Muga, S. Brouard, D. Macias, Time of arrival in quantum mechanics. Ann. Phys. 240, 351–366 (1995). https://doi.org/10.1006/aphy.1995.1048

    Article  ADS  MathSciNet  Google Scholar 

  14. P.C.W. Davies, Quantum mechanics and the equivalence principle. Class. Quantum Gravity 21, 2761–2772 (2004). https://doi.org/10.1088/0264-9381/21/11/017

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. P.C.W. Davies, Transit time of a freely falling quantum particle in a background gravitational field. Class. Quantum Gravity 21, 5677–5683 (2004). https://doi.org/10.1088/0264-9381/21/24/001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. H. Salecker, E.P. Wigner, Quantum limitations of the measurement of space-time distances. Phys. Rev. 109, 571–577 (1958). https://doi.org/10.1103/PhysRev.109.571

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. A. Peres, Measurement of time by quantum clocks. Am. J. Phys. 48, 552–557 (1980). https://doi.org/10.1119/1.12061

    Article  ADS  MathSciNet  Google Scholar 

  18. C. Anastopoulos, B.L. Hu, Equivalence principle for quantum systems: dephasing and phase shift of free-falling particles. Class. Quantum Gravity 35, 035011 (2018). https://doi.org/10.1088/1361-6382/aaa0e8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. J. Finkelstein, Ambiguities of arrival-time distributions in quantum theory. Phys. Rev. A 59, 3218–3222 (1999). https://doi.org/10.1103/PhysRevA.59.3218

    Article  ADS  MathSciNet  Google Scholar 

  20. J.G. Muga, C.R. Leavens, Arrival time in quantum mechanics. Phys. Rep. 338, 353–438 (2000). https://doi.org/10.1016/S0370-1573(00)00047-8

    Article  ADS  MathSciNet  Google Scholar 

  21. M. Moshinsky, Diffraction in time. Phys. Rev. 88, 625–631 (1952). https://doi.org/10.1103/PhysRev.88.625

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. P. Szriftgiser, D. Guéry-Odelin, M. Arndt, J. Dalibard, Atomic wave diffraction and interference using temporal slits. Phys. Rev. Lett. 77, 4–7 (1996). https://doi.org/10.1103/PhysRevLett.77.4

    Article  ADS  Google Scholar 

  23. A. del Campo, G. García-Calderón, J.G. Muga, Quantum transients. Phys. Rep. 476, 1–50 (2009). https://doi.org/10.1016/j.physrep.2009.03.002

    Article  ADS  Google Scholar 

  24. Saurya Das, Elias C. Vagenas, Universality of quantum gravity corrections. Phys. Rev. Lett. 101, 221301 (2008). https://doi.org/10.1103/PhysRevLett.101.221301

    Article  ADS  Google Scholar 

  25. A. Martín-Ruiz, Diffraction in time of polymer particles. Phys. Rev. D 90, 125027 (2014). https://doi.org/10.1103/PhysRevD.90.125027

    Article  ADS  Google Scholar 

  26. S. Longhi, Equivalence principle and quantum mechanics: quantum simulation with entangled photons. Opt. Lett. 43, 226–229 (2018). https://doi.org/10.1364/OL.43.000226

    Article  ADS  Google Scholar 

  27. R.P. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics (McGraw-Hill, New York, 1965)

    MATH  Google Scholar 

  28. I.S. Gradshteyn, I.M. Ryzhik, in Table of Integrals, Series, and Products, 4th edn., ed. by A. Jeffrey, D. Zwillinger (Academic Press, New York, 1994)

  29. W. Emrich, Chapter 5: Basic nuclear structure and processes, in Principles of Nuclear Rocket Propulsion. ed. by W. Emrich (Butterworth-Heinemann, Oxford, 2016), pp. 55–80

    Chapter  Google Scholar 

  30. V.V. Nesvizhevsky, H.G. Börner, A.K. Petukhov, H. Abele, S. Baeßler, F.J. Rueß, T. Stöferle, A. Westphal, A.M. Gagarski, G.A. Petrov, A.V. Strelkov, Quantum states of neutrons in the earth’s gravitational field. Nature 415, 297–299 (2002). https://doi.org/10.1038/415297a

    Article  ADS  Google Scholar 

  31. C.G. Aminoff, A.M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, C. Cohen-Tannoudji, Cesium atoms bouncing in a stable gravitational cavity. Phys. Rev. Lett. 71, 3083–3086 (1993). https://doi.org/10.1103/PhysRevLett.71.3083

    Article  ADS  Google Scholar 

  32. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, A. Zeilinger, Wave-particle duality of C60 molecules. Nature 401, 680–682 (1999). https://doi.org/10.1038/44348

    Article  ADS  Google Scholar 

  33. A. Goel, J.B. Howard, J.B. Vander Sande, Size analysis of single fullerene molecules by electron microscopy. Carbon 42, 1907–1915 (2004). https://doi.org/10.1016/j.carbon.2004.03.022

    Article  Google Scholar 

  34. R.W. Robinett, Quantum and classical probability distributions for position and momentum. Am. J. Phys. 63, 823–832 (1995). https://doi.org/10.1119/1.17807

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. G. Yoder, Using classical probability functions to illuminate the relation between classical and quantum physics. Am. J. Phys. 74, 404–411 (2006). https://doi.org/10.1119/1.2173280

    Article  ADS  Google Scholar 

  36. E.G.P. Rowe, The classical limit of quantum mechanical hydrogen radial distributions. Eur. J. Phys. 8, 81–87 (1987). https://doi.org/10.1088/0143-0807/8/2/002

    Article  Google Scholar 

  37. J. Bernal, A. Martín-Ruiz, J. García-Melgarejo, A simple mathematical formulation of the correspondence principle. J. Mod. Phys. 4, 108 (2013). https://doi.org/10.4236/jmp.2013.41017

    Article  Google Scholar 

  38. A. Martín-Ruiz, J. Bernal, A. Frank, A. Carbajal-Dominguez, The classical limit of the quantum Kepler problem. J. Mod. Phys. 4, 818 (2013). https://doi.org/10.4236/jmp.2013.46112

    Article  Google Scholar 

  39. A. Martín-Ruiz, J. Bernal, A. Carbajal-Dominguez, Macroscopic quantum behaviour of periodic quantum systems. J. Mod. Phys. 5, 44 (2013). https://doi.org/10.4236/jmp.2014.51007

    Article  Google Scholar 

  40. J.A. Cañas, J. Bernal, A. Martín-Ruiz, Exact classical limit of the quantum bouncer. Unpublished

  41. S. Baeßler, V.V. Nesvizhevsky, G. Pignol, K.V. Protasov, AYu. Voronin, Constraints on spin-dependent short-range interactions using gravitational quantum levels of ultracold neutrons. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 611, 149–152 (2009). https://doi.org/10.1016/j.nima.2009.07.048

    Article  ADS  Google Scholar 

  42. A. Martín-Ruiz, C.A. Escobar, Testing Lorentz and CPT invariance with ultracold neutrons. Phys. Rev. D 97, 095039 (2018). https://doi.org/10.1103/PhysRevD.97.095039

    Article  ADS  Google Scholar 

  43. C.A. Escobar, A. Martín-Ruiz, Gravitational searches for Lorentz violation with ultracold neutrons. Phys. Rev. D 99, 075032 (2019). https://doi.org/10.1103/PhysRevD.99.075032

    Article  ADS  Google Scholar 

  44. O. Vallée, M. Soares, Airy Functions and Applications to Physics (Imperial College Press, London, 2004)

    Book  Google Scholar 

  45. V.V. Nesvizhevsky, H. Börner, A.M. Gagarski, G.A. Petrov, A.K. Petukhov, H. Abele, S. Bäßler, T. Stöferle, S.M. Soloviev, Search for quantum states of the neutron in a gravitational field: gravitational levels. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 440, 754–759 (2000). https://doi.org/10.1016/S0168-9002(99)01077-3

    Article  ADS  Google Scholar 

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Acknowledgements

J.A.C. was supported by the CONACyT master fellowship No. 725033. A.M.-R. has been partially supported by DGAPA-UNAM Project No. IA102722 and by Project CONACyT (México) No. 428214. We thank to M. Cambiaso, M. J. Everitt and C. Escobar for their careful reading of the manuscript.

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Authors and Affiliations

  1. División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, 86690, Cunduacán, Tabasco, México

    Juan A. Cañas & J. Bernal

  2. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510, Ciudad de México, México

    A. Martín-Ruiz

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  1. Juan A. Cañas
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  2. J. Bernal
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Correspondence to A. Martín-Ruiz.

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Cañas, J.A., Bernal, J. & Martín-Ruiz, A. Testing the equivalence principle with time-diffracted free-falling quantum particles. Eur. Phys. J. Plus 137, 816 (2022). https://doi.org/10.1140/epjp/s13360-022-03051-5

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