Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Tests of fundamental quantum mechanics and dark interactions with low-energy neutrons

Abstract

Among the known particles, the neutron is special because it provides experimental access to all of the four fundamental forces and a wide range of hypothetical interactions. Despite being unstable, free neutrons live long enough to be used as test particles in interferometric, spectroscopic and scattering experiments probing low-energy scales. Recognized already in the 1970s, fundamental concepts of quantum mechanics can be tested in neutron interferometry using silicon perfect single crystals. Besides enabling tests of uncertainty relations or Bell inequalities, neutrons offer the opportunity to observe the effects of gravity and hypothetical dark forces acting on extended matter wavefunctions. Such tests gained importance in the light of recent discoveries of inconsistencies in the understanding of cosmology and the incompatibility between quantum mechanics and general relativity. Experiments with low-energy neutrons are, thus, indispensable tools for probing fundamental physics and represent a complementary approach to particle colliders. In this Review, we discuss the history and experimental methods used at this low-energy frontier of physics and overview the current bounds and limits on quantum mechanical relations and dark energy interactions.

Key points

  • The neutron is an excellent probe of various interactions, because it is sensitive to all four fundamental forces and hypothetical beyond-standard-model interactions.

  • Matter-wave interferometry with neutrons offers several advantages, such as macroscopic beam separation, individual control of the sub-beams and long interaction and coherence times at room temperature.

  • The neutron as a single-particle quantum system is almost ideal to study fundamental concepts of quantum mechanics, such as entanglement, weak values or uncertainty relations, because it can be prepared, manipulated and detected with high efficiency and accuracy.

  • Tensions between observation and the standard model of Big Bang cosmology and the ongoing quest to understand the nature of the dark sector are a strong motivation to search for new physics.

  • Neutrons, with their vanishing electric charge and large mass, are ideal test particles, allowing for the most sensitive measurements at sub-micrometre distances for several hypothetical dark sector models.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Entangled quantum state for photons (γ, multi-particle entanglement) in comparison with entangled single-neutron (n) quantum state (bipartite and tripartite intra-particle entanglement).
Fig. 2: Neutron interferometric procedure for weak measurements.
Fig. 3: Neutron polarimetric approaches for studying uncertainty relations.
Fig. 4: Compiled limits on non-Newtonian interactions.

References

  1. 1.

    De Broglie, L. V. Recherches sur la théorie des quanta [French]. English translation: J. W. Haslett. Am. J. Phys. 40, 1315–1320 (1972).

    Article  Google Scholar 

  2. 2.

    Bohr, N. The quantum postulate and the recent development of atomic theory. Nature 121, 580–590 (1928).

    ADS  MATH  Article  Google Scholar 

  3. 3.

    Abele, H. The neutron. Its properties and basic interactions. Prog. Part. Nucl. Phys. 60, 1–81 (2008).

    ADS  Article  Google Scholar 

  4. 4.

    Brax, P. & Pignol, G. Strongly coupled chameleons and the neutronic quantum bouncer. Phys. Rev. Lett. 107, 111301 (2011).

    ADS  Article  Google Scholar 

  5. 5.

    Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).

    ADS  MATH  Article  Google Scholar 

  6. 6.

    Shimony, A. in Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science, Bucharest, 1971 Vol. 74 of Studies in Logic and the Foundations of Mathematics (eds Suppes, P., Henkin, L., Joja, A. & Moisil, G. C.) 593–601 (Elsevier, 1973).

  7. 7.

    Bell, J. S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Mermin, N. D. Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Bell, J. S. Speakable and Unspeakable in Quantum Mechanics (Cambridge Univ. Press, 1987).

    MATH  Google Scholar 

  10. 10.

    Colella, R., Overhauser, A. W. & Werner, S. A. Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472–1474 (1975).

    ADS  Article  Google Scholar 

  11. 11.

    Werner, S. A., Staudenmann, J. L. & Colella, R. Effect of Earth’s rotation on the quantum mechanical phase of the neutron. Phys. Rev. Lett. 42, 1103–1106 (1979).

    ADS  Article  Google Scholar 

  12. 12.

    Arif, M. et al. Observation of a motion-induced phase shift of neutron de broglie waves passing through matter near a nuclear resonance. Phys. Rev. A 39, 931–937 (1989).

    ADS  Article  Google Scholar 

  13. 13.

    Danner, A. et al. Spin-rotation coupling observed in neutron interferometry. npj Quantum Information 6, 23 (2020).

    ADS  Article  Google Scholar 

  14. 14.

    Jenke, T., Geltenbort, P., Lemmel, H. & Abele, H. Realization of a gravity-resonance-spectroscopy technique. Nat. Phys. 7, 468–472 (2011).

    Article  Google Scholar 

  15. 15.

    Jenke, T. et al. Gravity resonance spectroscopy constrains dark energy and dark matter scenarios. Phys. Rev. Lett. 112, 151105 (2014).

    ADS  Article  Google Scholar 

  16. 16.

    Cronenberg, G. et al. Acoustic Rabi oscillations between gravitational quantum states and impact on symmetron dark energy. Nat. Phys. 14, 1022–1026 (2018).

    Article  Google Scholar 

  17. 17.

    Lemmel, H. et al. Neutron interferometry constrains dark energy chameleon fields. Phys. Lett. B 743, 310–314 (2015).

    ADS  Article  Google Scholar 

  18. 18.

    Möllenstedt, G. & Düker, H. Electron interferometer. Z. Phys. 145, 377 (1956).

    ADS  Article  Google Scholar 

  19. 19.

    Rauch, H., Treimer, W. & Bonse, U. Test of a single crystal neutron interferometer. Phys. Lett. A 47, 369–371 (1974).

    ADS  Article  Google Scholar 

  20. 20.

    Rauch, H. & Werner, S. A. Neutron Interferometry (Clarendon, 2000).

    Google Scholar 

  21. 21.

    Carnal, O. & Mlynek, J. Young’s double-slit experiment with atoms: a simple atom interferometer. Phys. Rev. Lett. 66, 2689–2692 (1991).

    ADS  Article  Google Scholar 

  22. 22.

    Keith, D. W., Ekstrom, C. R., Turchette, Q. A. & Pritchard, D. E. An interferometer for atoms. Phys. Rev. Lett. 66, 2693–2696 (1991).

    ADS  Article  Google Scholar 

  23. 23.

    Kasevich, M. & Chu, S. Atomic interferometry using stimulated Raman transitions. Phys. Rev. Lett. 67, 181–184 (1991).

    ADS  Article  Google Scholar 

  24. 24.

    Riehle, F., Kisters, T., Witte, A., Helmcke, J. & Bordé, C. J. Optical Ramsey spectroscopy in a rotating frame: Sagnac effect in a matter-wave interferometer. Phys. Rev. Lett. 67, 177–180 (1991).

    ADS  Article  Google Scholar 

  25. 25.

    Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M. & Tüxen, J. Matter–wave interference of particles selected from a molecular library with masses exceeding 10000 amu. Phys. Chem. Chem. Phys. 15, 14696–14700 (2013).

    Article  Google Scholar 

  26. 26.

    Hasegawa, Y., Loidl, R., Badurek, G., Baron, M. & Rauch, H. Violation of a Bell-like inequality in single-neutron interferometry. Nature 425, 45–48 (2003).

    ADS  MATH  Article  Google Scholar 

  27. 27.

    Sponar, S. et al. Geometric phase in entangled systems: a single-neutron interferometer experiment. Phys. Rev. A 81, 042113 (2010).

    ADS  Article  Google Scholar 

  28. 28.

    Geppert, H., Denkmayr, T., Sponar, S., Lemmel, H. & Hasegawa, Y. Improvement of the polarized neutron interferometer setup demonstrating violation of a Bell-like inequality. Nucl. Instrum. Methods Phys. Res. A 763, 417–423 (2014).

    ADS  Article  Google Scholar 

  29. 29.

    Klepp, J., Sponar, S. & Hasegawa, Y. Fundamental phenomena of quantum mechanics explored with neutron interferometers. Prog. Theor. Exp. Phys. 2014, 082A0 (2014).

    Article  Google Scholar 

  30. 30.

    Hasegawa, Y. et al. Engineering of triply entangled states in a single-neutron system. Phys. Rev. A 81, 032121 (2010).

    ADS  Article  Google Scholar 

  31. 31.

    Erdösi, D., Huber, M., Hiesmayr, B. C. & Hasegawa, Y. Proving the generation of genuine multipartite entanglement in a single-neutron interferometer experiment. New J. Phys. 15, 023033 (2013).

    ADS  Article  Google Scholar 

  32. 32.

    Hasegawa, Y., Loidl, R., Badurek, G., Baron, M. & Rauch, H. Quantum contextuality in a single-neutron optical experiment. Phys. Rev. Lett. 97, 230401 (2006).

    ADS  MATH  Article  Google Scholar 

  33. 33.

    Bartosik, H. et al. Experimental test of quantum contextuality in neutron interferometry. Phys. Rev. Lett. 103, 040403 (2009).

    ADS  Article  Google Scholar 

  34. 34.

    Denkmayr, T. et al. Experimental observation of a quantum Cheshire Cat in matter-wave interferometry. Nat. Commun. 5, 4492 (2014).

    ADS  Article  Google Scholar 

  35. 35.

    Sponar, S. et al. Weak values obtained in matter-wave interferometry. Phys. Rev. A 92, 062121 (2015).

    ADS  Article  Google Scholar 

  36. 36.

    Denkmayr, T. et al. Experimental demonstration of direct path state characterization by strongly measuring weak values in a matter-wave interferometer. Phys. Rev. Lett. 118, 010402 (2017).

    ADS  Article  Google Scholar 

  37. 37.

    Geppert-Kleinrath, H. et al. Multifold paths of neutrons in the three-beam interferometer detected by a tiny energy kick. Phys. Rev. A 97, 052111 (2018).

    ADS  Article  Google Scholar 

  38. 38.

    Cronin, A. D., Schmiedmayer, J. & Pritchard, D. E. Optics and interferometry with atoms and molecules. Rev. Mod. Phys. 81, 1051–1129 (2009).

    ADS  Article  Google Scholar 

  39. 39.

    Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, J. Local emergence of thermal correlations in an isolated quantum many-body system. Nat. Phys. 9, 640–643 (2013).

    Article  Google Scholar 

  40. 40.

    Babb, J. F. & Hussein, M. S. Van der Waals and Casimir-Polder interactions between neutrons. EPJ Web Conf. 113, 08001 (2016).

    Article  Google Scholar 

  41. 41.

    Hasegawa, Y. & Rauch, H. Quantum phenomena explored with neutrons. New J. Phys. 13, 115010 (2011).

    ADS  Article  Google Scholar 

  42. 42.

    Huber, M. G. et al. Overview of neutron interferometry at NIST. EPJ Web Conf. 219, 06001 (2019).

    Article  Google Scholar 

  43. 43.

    Weiss, D. S., Young, B. C. & Chu, S. Precision measurement of the photon recoil of an atom using atomic interferometry. Phys. Rev. Lett. 70, 2706–2709 (1993).

    ADS  Article  Google Scholar 

  44. 44.

    Andreas, B. et al. Determination of the Avogadro constant by counting the atoms in a 28Si crystal. Phys. Rev. Lett. 106, 030801 (2011).

    ADS  Article  Google Scholar 

  45. 45.

    Parker, R. H., Yu, C., Zhong, W., Estey, B. & Müller, H. Measurement of the fine-structure constant as a test of the standard model. Science 360, 191–195 (2018).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Rosi, G., Sorrentino, F., Cacciapuoti, L., Prevedelli, M. & Tino, G. M. Precision measurement of the Newtonian gravitational constant using cold atoms. Nature 510, 518–521 (2014).

    ADS  Article  Google Scholar 

  47. 47.

    Schlippert, D. et al. Quantum test of the universality of free fall. Phys. Rev. Lett. 112, 203002 (2014).

    ADS  Article  Google Scholar 

  48. 48.

    Hamilton, P. et al. Atom-interferometry constraints on dark energy. Science 349, 849–851 (2015).

    ADS  Article  Google Scholar 

  49. 49.

    Jaffe, M. et al. Testing sub-gravitational forces on atoms from a miniature in-vacuum source mass. Nat. Phys. 13, 938 (2017).

    Article  Google Scholar 

  50. 50.

    van Zoest, T. et al. Bose-Einstein condensation in microgravity. Science 328, 1540–1543 (2010).

    ADS  Article  Google Scholar 

  51. 51.

    Müntinga, H. et al. Interferometry with Bose-Einstein condensates in microgravity. Phys. Rev. Lett. 110, 093602 (2013).

    ADS  Article  Google Scholar 

  52. 52.

    Becker, D. et al. Space-borne Bose–Einstein condensation for precision interferometry. Nature 562, 391–395 (2018).

    ADS  Article  Google Scholar 

  53. 53.

    Elliott, E. R., Krutzik, M. C., Williams, J. R., Thompson, R. J. & Aveline, D. C. NASA’s Cold Atom Lab (CAL): system development and ground test status. npj Microgravity 4, 16 (2018).

    ADS  Article  Google Scholar 

  54. 54.

    Bongs, K. et al. Taking atom interferometric quantum sensors from the laboratory to real-world applications. Nat. Rev. Phys. 1, 731–739 (2019).

    Article  Google Scholar 

  55. 55.

    Bertone, G. (ed.) Particle Dark Matter: Observations, Models and Searches (Cambridge Univ. Press, 2010).

  56. 56.

    Weinberg, S. A new light boson? Phys. Rev. Lett. 40, 223–226 (1978).

    ADS  Article  Google Scholar 

  57. 57.

    Wilczek, F. Problem of strong P and T invariance in the presence of instantons. Phys. Rev. Lett. 40, 279–282 (1978).

    ADS  Article  Google Scholar 

  58. 58.

    Kim, J. E. Weak-interaction singlet and CP invariance. Phys. Rev. Lett. 43, 103–107 (1979).

    ADS  Article  Google Scholar 

  59. 59.

    Shifman, M. A., Vainshtein, A. I. & Zakharov, V. I. Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B 166, 493–506 (1980).

    ADS  MathSciNet  Article  Google Scholar 

  60. 60.

    Dine, M., Fischler, W. & Srednicki, M. A simple solution to the strong CP problem with a harmless axion. Phys. Lett. B 104, 199–202 (1981).

    ADS  Article  Google Scholar 

  61. 61.

    Peccei, R. D. & Quinn, H. R. CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440–1443 (1977).

    ADS  Article  Google Scholar 

  62. 62.

    Peccei, R. D. & Quinn, H. R. Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D 16, 1791–1797 (1977).

    ADS  Article  Google Scholar 

  63. 63.

    Raffelt, G. G. Particle physics from stars. Annu. Rev. Nucl. Part. Sci. 49, 163–216 (1999).

    ADS  Article  Google Scholar 

  64. 64.

    Moody, J. E. & Wilczek, F. New macroscopic forces? Phys. Rev. D 30, 130–138 (1984).

    ADS  Article  Google Scholar 

  65. 65.

    Freivogel, B. Anthropic explanation of the dark matter abundance. J. Cosmol. Astropart. Phys. 2010, 021 (2010).

    Article  Google Scholar 

  66. 66.

    Linde, A. D. Inflation and axion cosmology. Phys. Lett. B 201, 437–439 (1988).

    ADS  Article  Google Scholar 

  67. 67.

    Duffy, L. D. & van Bibber, K. Axions as dark matter particles. New J. Phys. 11, 105008 (2009).

    ADS  Article  Google Scholar 

  68. 68.

    Graham, P. W., Irastorza, I. G., Lamoreaux, S. K., Lindner, A. & van Bibber, K. A. Experimental searches for the axion and axion-like particles. Annu. Rev. Nucl. Part. Sci. 65, 485–514 (2015).

    ADS  Article  Google Scholar 

  69. 69.

    Mantry, S., Pitschmann, M. & Ramsey-Musolf, M. J. Distinguishing axions from generic light scalars using electric dipole moment and fifth-force experiments. Phys. Rev. D 90, 054016 (2014).

    ADS  Article  Google Scholar 

  70. 70.

    Perlmutter, S. et al. Discovery of a supernova explosion at half the age of the universe. Nature 391, 51–54 (1998).

    ADS  Article  Google Scholar 

  71. 71.

    Riess, A. G. et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998).

    ADS  Article  Google Scholar 

  72. 72.

    Schmidt, B. P. et al. The high-Z supernova search: measuring cosmic deceleration and global curvature of the universe using type Ia supernovae. Astrophys. J. 507, 46 (1998).

    ADS  Article  Google Scholar 

  73. 73.

    Einstein, A. Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsberichte K. Preußischen Akad. Wiss. Berl. 142–152 (1917).

  74. 74.

    Solà, J. Cosmological constant and vacuum energy: old and new ideas. J. Phys. Conf. Ser. 453, 012015 (2013).

    Article  Google Scholar 

  75. 75.

    Joyce, A., Jain, B., Khoury, J. & Trodden, M. Beyond the cosmological standard model. Phys. Rep. 568, 1–98 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  76. 76.

    Will, C. M. The confrontation between general relativity and experiment. Living Rev. Relativ. 17, 4 (2014).

    ADS  MATH  Article  Google Scholar 

  77. 77.

    Khoury, J. & Weltman, A. Chameleon cosmology. Phys. Rev. D 69, 044026 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  78. 78.

    Khoury, J. & Weltman, A. Chameleon fields: awaiting surprises for tests of gravity in space. Phys. Rev. Lett. 93, 171104 (2004).

    ADS  Article  Google Scholar 

  79. 79.

    Mota, D. F. & Shaw, D. J. Strongly coupled chameleon fields: new horizons in scalar field theory. Phys. Rev. Lett. 97, 151102 (2006).

    ADS  Article  Google Scholar 

  80. 80.

    Mota, D. F. & Shaw, D. J. Evading equivalence principle violations, cosmological, and other experimental constraints in scalar field theories with a strong coupling to matter. Phys. Rev. D 75, 063501 (2007).

    ADS  Article  Google Scholar 

  81. 81.

    Waterhouse, T. P. An introduction to chameleon gravity. Preprint at arXiv astro-ph/0611816 (2006).

  82. 82.

    Hinterbichler, K. & Khoury, J. Screening long-range forces through local symmetry restoration. Phys. Rev. Lett. 104, 231301 (2010).

    ADS  Article  Google Scholar 

  83. 83.

    Hinterbichler, K., Khoury, J., Levy, A. & Matas, A. Symmetron cosmology. Phys. Rev. D 84, 103521 (2011).

    ADS  Article  Google Scholar 

  84. 84.

    Pietroni, M. Dark energy condensation. Phys. Rev. D 72, 043535 (2005).

    ADS  Article  Google Scholar 

  85. 85.

    Olive, K. A. & Pospelov, M. Environmental dependence of masses and coupling constants. Phys. Rev. D 77, 043524 (2008).

    ADS  Article  Google Scholar 

  86. 86.

    Rauch, H. et al. Verification of coherent spinor rotation of fermions. Phys. Lett. A 54, 425–427 (1975).

    ADS  Article  Google Scholar 

  87. 87.

    Summhammer, J., Badurek, G., Rauch, H., Kischko, U. & Zeilinger, A. Direct observation of fermion spin superposition by neutron interferometry. Phys. Rev. A 27, 2523–2532 (1983).

    ADS  Article  Google Scholar 

  88. 88.

    Badurek, G., Rauch, H. & Summhammer, J. Time-dependent superposition of spinors. Phys. Rev. Lett. 51, 1015–1018 (1983).

    ADS  Article  Google Scholar 

  89. 89.

    Badurek, G., Rauch, H. & Tuppinger, D. Neutron interferometric double-resonance experiment. Phys. Rev. A 34, 2600–2608 (1986).

    ADS  Article  Google Scholar 

  90. 90.

    Rauch, H., Lemmel, H., Baron, M. & Loidl, R. Measurement of a confinement induced neutron phase. Nature 417, 630–632 (2002).

    ADS  Article  Google Scholar 

  91. 91.

    Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).

    ADS  MATH  Article  Google Scholar 

  92. 92.

    Bell, J. S. On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964).

    MathSciNet  Article  Google Scholar 

  93. 93.

    Kochen, S. & Specker, E. P. The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967).

    MathSciNet  MATH  Google Scholar 

  94. 94.

    Hasegawa, Y., Loidl, R., Badurek, G., Baron, M. & Rauch, H. Quantum contextuality in a single-neutron optical experiment. Phys. Rev. Lett. 97, 230401 (2006).

    ADS  MATH  Article  Google Scholar 

  95. 95.

    Cabello, A., Filipp, S., Rauch, H. & Hasegawa, Y. Proposed experiment for testing quantum contextuality with neutrons. Phys. Rev. Lett. 100, 130404 (2008).

    ADS  Article  Google Scholar 

  96. 96.

    Greenberger, D. M., Horne, M. A. & Zeilinger in Kafatos, M. (ed.) Bells Theorem, Quantum Theory, and Concepts of the Universe (ed. Kafatos, M.) 73–76 (Kluwer, 1989).

  97. 97.

    Greenberger, D. M., Shimony, A., Horne, M. A. & Zeilinger, A. Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  98. 98.

    Mermin, N. D. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373–3376 (1990).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  99. 99.

    Shen, J. et al. Unveiling contextual realities by microscopically entangling a neutron. Nat. Commun. 11, 930 (2020).

    ADS  Article  Google Scholar 

  100. 100.

    Verbeeck, J., Tian, H. & Schattschneider, P. Production and application of electron vortex beams. Nature 476, 301–304 (2010).

    ADS  Article  Google Scholar 

  101. 101.

    Clark, C. W. et al. Controlling neutron orbital angular momentum. Nature 525, 504–507 (2015).

    ADS  Article  Google Scholar 

  102. 102.

    Sarenac, D. et al. Generation and detection of spin-orbit coupled neutron beams. Proc. Natl Acad. Sci. USA 116, 20328–20332 (2019).

    ADS  Article  Google Scholar 

  103. 103.

    Nsofini, J. et al. Spin-orbit states of neutron wave packets. Phys. Rev. A 94, 013605 (2016).

    ADS  Article  Google Scholar 

  104. 104.

    Heckel, B. et al. Measurement of parity nonconserving neutron spin rotation in lanthanum. Phys. Rev. C 29, 2389–2391 (1984).

    ADS  Article  Google Scholar 

  105. 105.

    Geerits, N. & Sponar, S. Twisting neutral particles with electric fields. Phys. Rev. A 103, 022205 (2021).

    ADS  Article  Google Scholar 

  106. 106.

    Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988).

    ADS  Article  Google Scholar 

  107. 107.

    Ritchie, N. W. M., Story, J. G. & Hulet, R. G. Realization of a measurement of a “weak value”. Phys. Rev. Lett. 66, 1107–1110 (1991).

    ADS  Article  Google Scholar 

  108. 108.

    Hosten, O. & Kwiat, P. Observation of the spin Hall effect of light via weak measurements. Science 319, 787–790 (2008).

    ADS  Article  Google Scholar 

  109. 109.

    Dixon, P. B., Starling, D. J., Jordan, A. N. & Howell, J. C. Ultrasensitive beam deflection measurement via interferometric weak value amplification. Phys. Rev. Lett. 102, 173601 (2009).

    ADS  Article  Google Scholar 

  110. 110.

    Starling, D. J., Dixon, P. B., Jordan, A. N. & Howell, J. C. Precision frequency measurements with interferometric weak values. Phys. Rev. A 82, 063822 (2010).

    ADS  Article  Google Scholar 

  111. 111.

    Starling, D. J., Dixon, P. B., Williams, N. S., Jordan, A. N. & Howell, J. C. Continuous phase amplification with a Sagnac interferometer. Phys. Rev. A 82, 011802 (2010).

    ADS  Article  Google Scholar 

  112. 112.

    Feizpour, A., Xing, X. & Steinberg, A. M. Amplifying single-photon nonlinearity using weak measurements. Phys. Rev. Lett. 107, 133603 (2011).

    ADS  Article  Google Scholar 

  113. 113.

    Zhou, L., Turek, Y., Sun, C. P. & Nori, F. Weak-value amplification of light deflection by a dark atomic ensemble. Phys. Rev. A 88, 053815 (2013).

    ADS  Article  Google Scholar 

  114. 114.

    Rozema, L. A. et al. Violation of Heisenberg’s measurement-disturbance relationship by weak measurements. Phys. Rev. Lett. 109, 100404 (2012).

    ADS  Article  Google Scholar 

  115. 115.

    Lundeen, J. S., Sutherland, B., Patel, A., Stewart, C. & Bamber, C. Direct measurement of the quantum wavefunction. Nature 474, 188–191 (2011).

    Article  Google Scholar 

  116. 116.

    Goggin, M. E. et al. Violation of the Leggett–Garg inequality with weak measurements of photons. Proc. Natl Acad. Sci. USA 108, 1256–1261 (2011).

    ADS  Article  Google Scholar 

  117. 117.

    Salvail, J. Z. et al. Full characterization of polarization states of light via direct measurement. Nat. Photonics 7, 316–321 (2013).

    ADS  Article  Google Scholar 

  118. 118.

    Kaneda, F., Baek, S.-Y., Ozawa, M. & Edamatsu, K. Experimental test of error-disturbance uncertainty relations by weak measurement. Phys. Rev. Lett. 112, 020402 (2014).

    ADS  Article  Google Scholar 

  119. 119.

    Ringbauer, M. et al. Experimental joint quantum measurements with minimum uncertainty. Phys. Rev. Lett. 112, 020401 (2014).

    ADS  Article  Google Scholar 

  120. 120.

    Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W. Colloquium: understanding quantum weak values: basics and applications. Rev. Mod. Phys. 86, 307–316 (2014).

    ADS  Article  Google Scholar 

  121. 121.

    Sulyok, G. & Sponar, S. Heisenberg’s error-disturbance uncertainty relation: experimental study of competing approaches. Phys. Rev. A 96, 022137 (2017).

    ADS  Article  Google Scholar 

  122. 122.

    Resch, J., Lundeen, J. & Steinberg, A. Experimental realization of the quantum box problem. Phys. Lett. A 324, 125–131 (2004).

    ADS  MATH  Article  Google Scholar 

  123. 123.

    Lundeen, J. S. & Steinberg, A. M. Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox. Phys. Rev. Lett. 102, 020404 (2009).

    ADS  Article  Google Scholar 

  124. 124.

    Yokota, K., Yamamoto, T., Koashi, M. & Imoto, N. Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair. New J. Phys. 11, 033011 (2009).

    ADS  Article  Google Scholar 

  125. 125.

    Aharonov, Y., Botero, A., Popescu, S., Reznik, B. & Tollaksen, J. Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A 301, 130–138 (2002).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  126. 126.

    Waegell, M. et al. Confined contextuality in neutron interferometry: observing the quantum pigeonhole effect. Phys. Rev. A 96, 052131 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  127. 127.

    Aharonov, Y., Popescu, S., Rohrlich, D. & Skrzypczyk, P. Quantum cheshire cats. New J. Phys. 15, 113015 (2013).

    ADS  MATH  Article  Google Scholar 

  128. 128.

    Ashby, J. M., Schwarz, P. D. & Schlosshauer, M. Observation of the quantum paradox of separation of a single photon from one of its properties. Phys. Rev. A 94, 012102 (2016).

    ADS  Article  Google Scholar 

  129. 129.

    Atherton, D. P., Ranjit, G., Geraci, A. A. & Weinstein, J. D. Observation of a classical Cheshire cat in an optical interferometer. Opt. Lett. 40, 879–881 (2015).

    ADS  Article  Google Scholar 

  130. 130.

    Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927).

    ADS  MATH  Article  Google Scholar 

  131. 131.

    Kennard, E. H. Zur quantenmechanik einfacher bewegungstypen. Z. Phys. 44, 326–352 (1927).

    ADS  MATH  Article  Google Scholar 

  132. 132.

    Robertson, H. P. The uncertainty principle. Phys. Rev. 34, 163–164 (1929).

    ADS  Article  Google Scholar 

  133. 133.

    Arthurs, E. & Goodman, M. S. Quantum correlations: a generalized Heisenberg uncertainty relation. Phys. Rev. Lett. 60, 2447–2449 (1988).

    ADS  MathSciNet  Article  Google Scholar 

  134. 134.

    Ozawa, M. Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A 67, 042105 (2003).

    ADS  Article  Google Scholar 

  135. 135.

    Ozawa, M. Uncertainty relations for noise and disturbance in generalized quantum measurements. Ann. Phys. 311, 350–416 (2004).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  136. 136.

    Branciard, C. Error-tradeoff and error-disturbance relations for incompatible quantum measurements. Proc. Natl Acad. Sci. USA 17, 6742–6747 (2013).

    ADS  Article  Google Scholar 

  137. 137.

    Busch, P., Lahti, P. & Werner, R. F. Proof of Heisenberg’s error-disturbance relation. Phys. Rev. Lett. 111, 160405 (2013).

    ADS  Article  Google Scholar 

  138. 138.

    Busch, P., Lahti, P. & Werner, R. F. Colloquium: quantum root-mean-square error and measurement uncertainty relations. Rev. Mod. Phys. 86, 1261–1281 (2014).

    ADS  Article  Google Scholar 

  139. 139.

    Ozawa, M. Physical content of Heisenberg’s uncertainty relation: limitation and reformulation. Phys. Lett. A 318, 21–29 (2003).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  140. 140.

    Hall, M. J. W. Prior information: how to circumvent the standard joint-measurement uncertainty relation. Phys. Rev. A 69, 052113 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  141. 141.

    Erhart, J. et al. Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurements. Nat. Phys. 8, 185–189 (2012).

    Article  Google Scholar 

  142. 142.

    Sulyok, G. et al. Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements. Phys. Rev. A 88, 022110 (2013).

    ADS  Article  Google Scholar 

  143. 143.

    Busch, P., Lahti, P. & Werner, R. F. Heisenberg uncertainty for qubit measurements. Phys. Rev. A 89, 012129 (2014).

    ADS  Article  Google Scholar 

  144. 144.

    Buscemi, F., Hall, M. J., Ozawa, M. & Wilde, M. M. Noise and disturbance in quantum measurements: an information-theoretic approach. Phys. Rev. Lett. 112, 050401 (2014).

    ADS  Article  Google Scholar 

  145. 145.

    Sulyok, G. et al. Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements. Phys. Rev. Lett. 115, 030401 (2015).

    ADS  Article  Google Scholar 

  146. 146.

    Ma, W. et al. Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances. Phys. Rev. Lett. 116, 160405 (2016).

    ADS  Article  Google Scholar 

  147. 147.

    Barchielli, A., Gregoratti, M. & Toigo, A. Measurement uncertainty relations for discrete observables: Relative entropy formulation. Commun. Math. Phys. 357, 1253–1304 (2018).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  148. 148.

    Mao, Y.-L. et al. Error-disturbance trade-off in sequential quantum measurements. Phys. Rev. Lett. 122, 090404 (2019).

    ADS  Article  Google Scholar 

  149. 149.

    Baek, S.-Y., Kaneda, F., Ozawa, M. & Edamatsu, K. Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation. Sci. Rep. 3, 2221 (2013).

    Article  Google Scholar 

  150. 150.

    Demirel, B., Sponar, S., Sulyok, G., Ozawa, M. & Hasegawa, Y. Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states. Phys. Rev. Lett. 117, 140402 (2016).

    ADS  Article  Google Scholar 

  151. 151.

    Abbott, A. A., Alzieu, P.-L., Hall, M. J. W. & Branciard, C. Tight state-independent uncertainty relations for qubit. Mathematics 4, 8 (2016).

    MATH  Article  Google Scholar 

  152. 152.

    Sponar, S., Danner, A., Obigane, K., Hack, S. & Hasegawa, Y. Experimental test of tight state-independent preparation uncertainty relations for qubits. Phys. Rev. A 102, 042204 (2020).

    ADS  Article  Google Scholar 

  153. 153.

    Rauch, H. & Waschkowski, W. in Low Energy Neutrons and Their Interaction with Nuclei and Matter. Part 1 Vol. 16A1 (ed. Schopper, H.) 1–29 (2000).

  154. 154.

    Willis, B. T. M., & Carlile, C. J. Experimental Neutron Scattering (Oxford Univ. Press, 2009).

  155. 155.

    Haddock, C. C. et al. Search for deviations from the inverse square law of gravity at nm range using a pulsed neutron beam. Phys. Rev. D 97, 062002 (2018).

    ADS  Article  Google Scholar 

  156. 156.

    Voronin, V. V., Kuznetsov, I. A. & Shapiro, D. D. Search for novel short-range forces between elementary particles in neutron scattering. JETP Lett. 107, 1–3 (2018).

    ADS  Article  Google Scholar 

  157. 157.

    Pokotilovski, Y. N. Constraints on new interactions from neutron scattering experiments. Phys. Atom. Nuclei 69, 924–931 (2006).

    ADS  Article  Google Scholar 

  158. 158.

    Sears, V. F. Electromagnetic neutron-atom interactions. Phys. Rep. 141, 281–317 (1986).

    ADS  Article  Google Scholar 

  159. 159.

    Leeb, H. & Schmiedmayer, J. Constraint on hypothetical light interacting bosons from low-energy neutron experiments. Phys. Rev. Lett. 68, 1472–1475 (1992).

    ADS  Article  Google Scholar 

  160. 160.

    Nesvizhevsky, V. V., Pignol, G. & Protasov, K. V. Neutron scattering and extra-short-range interactions. Phys. Rev. D 77, 034020 (2008).

    ADS  Article  Google Scholar 

  161. 161.

    Kamiya, Y., Itagaki, K., Tani, M., Kim, G. N. & Komamiya, S. Constraints on new gravitylike forces in the nanometer range. Phys. Rev. Lett. 114, 161101 (2015).

    ADS  Article  Google Scholar 

  162. 162.

    Sears, V. F. Fundamental aspects of neutron optics. Phys. Rep. 82, 1–29 (1982).

    ADS  Article  Google Scholar 

  163. 163.

    Maier-Leibnitz, H. Neutronenrefraktometer zur Absolutbestimmung kohärenter Streuquerschnitte. Z. Angew. Phys. 14, 738 (1962).

    Google Scholar 

  164. 164.

    Koester, L. Absolutmessung der kohärenten Streulänge von Quecksilber mit dem Neutronen-Schwerkraft-Refraktometer am FRM. Z. Phys. 182, 328–336 (1965).

    ADS  Article  Google Scholar 

  165. 165.

    Warner, M. & Gubernatis, J. E. Neutron refractive index: A Fermi-Huygens theory. Phys. Rev. B 32, 6347–6357 (1985).

    ADS  Article  Google Scholar 

  166. 166.

    Sears, V. F. Local-field refinement of neutron scattering lengths. Z. Phys. A 321, 443–449 (1985).

    ADS  Article  Google Scholar 

  167. 167.

    Goldberger, M. L. & Seitz, F. Theory of the refraction and the diffraction of neutrons by crystals. Phys. Rev. 71, 294–310 (1947).

    ADS  MATH  Article  Google Scholar 

  168. 168.

    Schmiedmayer, J., Riehs, P., Harvey, J. A. & Hill, N. W. Measurement of the electric polarizability of the neutron. Phys. Rev. Lett. 66, 1015–1018 (1991).

    ADS  Article  Google Scholar 

  169. 169.

    Overhauser, A. W. & Colella, R. Experimental test of gravitationally induced quantum interference. Phys. Rev. Lett. 33, 1237–1239 (1974).

    ADS  Article  Google Scholar 

  170. 170.

    Colella, R., Overhauser, A. W. & Werner, S. A. Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472–1474 (1975).

    ADS  Article  Google Scholar 

  171. 171.

    Staudenmann, J. L., Werner, S. A., Colella, R. & Overhauser, A. W. Gravity and inertia in quantum mechanics. Phys. Rev. A 21, 1419–1438 (1980).

    ADS  Article  Google Scholar 

  172. 172.

    Littrell, K. C., Allman, B. E. & Werner, S. A. Two-wavelength-difference measurement of gravitationally induced quantum interference phases. Phys. Rev. A 56, 1767–1780 (1997).

    ADS  Article  Google Scholar 

  173. 173.

    van der Zouw, G. et al. Aharonov–Bohm and gravity experiments with the very-cold-neutron interferometer. Nucl. Instrum. Methods Phys. Res. A 440, 568–574 (2000).

    ADS  Article  Google Scholar 

  174. 174.

    Parnell, S. R. et al. Search for exotic spin-dependent couplings of the neutron with matter using spin-echo based neutron interferometry. Phys. Rev. D 101, 122002 (2020).

    ADS  Article  Google Scholar 

  175. 175.

    Brax, P., Pignol, G. & Roulier, D. Probing strongly coupled chameleons with slow neutrons. Phys. Rev. D 88, 083004 (2013).

    ADS  Article  Google Scholar 

  176. 176.

    Mashhoon, B. Neutron interferometry in a rotating frame of reference. Phys. Rev. Lett. 61, 2639–2642 (1988).

    ADS  Article  Google Scholar 

  177. 177.

    Zel’dovic, Y. B. Storage of cold neutrons. J. Exp. Theor. Phys. 36, 1952f (1959).

    Google Scholar 

  178. 178.

    Lushchikov, V. I., Pokotilovskii, Y. N., Strelkov, A. V. & Shapiro, F. L. Observation of ultracold neutrons. JETP Lett. 9, 40–45 (1969).

    Google Scholar 

  179. 179.

    Steyerl, A. Measurements of total cross sections for very slow neutrons with velocities from 100 m/sec to 5 m/sec. Phys. Lett. B 29, 33–35 (1969).

    ADS  Article  Google Scholar 

  180. 180.

    Gibbs, R. L. The quantum bouncer. Am. J. Phys. 43, 25–28 (1975).

    ADS  Article  Google Scholar 

  181. 181.

    Gea-Banacloche, J. A quantum bouncing ball. Am. J. Phys. 67, 776–782 (1999).

    ADS  Article  Google Scholar 

  182. 182.

    Nesvizhevsky, V. V. et al. Quantum states of neutrons in the Earth’s gravitational field. Nature 415, 297–299 (2002).

    ADS  Article  Google Scholar 

  183. 183.

    Nesvizhevsky, V. V. et al. Measurement of quantum states of neutrons in the Earth’s gravitational field. Phys. Rev. D 67, 102002 (2003).

    ADS  Article  Google Scholar 

  184. 184.

    Nesvizhevsky, V. V. et al. Study of the neutron quantum states in the gravity field. Eur. Phys. J. C 40, 479–491 (2005).

    ADS  Article  Google Scholar 

  185. 185.

    Westphal, A. et al. A quantum mechanical description of the experiment on the observation of gravitationally bound states. Eur. Phys. J. C 51, 367–375 (2007).

    ADS  Article  Google Scholar 

  186. 186.

    Abele, H., Baeßler, S. & Westphal, A. in Quantum Gravity: from Theory to Experimental Search, Lecture Notes in Physics (eds Giulini, D. J. W., Kiefer, C. & Lämmerzahl, C.) 355–366 (Springer, 2003).

  187. 187.

    Nesvizhevsky, V. V. & Protasov, K. V. Constraints on non-Newtonian gravity from the experiment on neutron quantum states in the earth’s gravitational field. Class. Quantum Grav. 21, 4557–4566 (2004).

    ADS  MATH  Article  Google Scholar 

  188. 188.

    Zimmer, O. & Kaiser, N. Comment about constraints on nanometer-range modifications to gravity from low-energy neutron experiments. Class. Quantum Grav. 23, 6077–6080 (2006).

    ADS  MATH  Article  Google Scholar 

  189. 189.

    Abel, C. et al. Measurement of the permanent electric dipole moment of the neutron. Phys. Rev. Lett. 124, 081803 (2020).

    ADS  Article  Google Scholar 

  190. 190.

    Kreuz, M. et al. A method to measure the resonance transitions between the gravitationally bound quantum states of neutrons in the GRANIT spectrometer. Nucl. Instrum. Methods Phys. Res. A 611, 326–330 (2009).

    ADS  Article  Google Scholar 

  191. 191.

    Pignol, G. et al. Gravitational resonance spectroscopy with an oscillating magnetic field gradient in the GRANIT flow through arrangement. Adv. High Energy Phys. 2014, e628125 (2014).

    Article  Google Scholar 

  192. 192.

    Abele, H., Ivanov, A., Jenke, T., Pitschmann, M. & Geltenbort, P. Gravity resonance spectroscopy and Einstein-Cartan gravity. 11th Patras Workshop on Axions, WIMPs and WISPs. Preprint at arXiv 1510.03063 (2015).

  193. 193.

    Ivanov, A. N. & Wellenzohn, M. Spin precession of slow neutrons in Einstein-Cartan gravity with torsion, chameleon, and magnetic field. Phys. Rev. D 93, 045031 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  194. 194.

    Durstberger-Rennhofer, K., Jenke, T. & Abele, H. Probing the neutron’s electric neutrality with Ramsey spectroscopy of gravitational quantum states of ultracold neutrons. Phys. Rev. D 84, 036004 (2011).

    ADS  Article  Google Scholar 

  195. 195.

    Sedmik, R. I. P. et al. Proof of principle for Ramsey-type gravity resonance spectroscopy with qBounce. EPJ Web Conf. 219, 05004 (2019).

    Article  Google Scholar 

  196. 196.

    Abele, H., Jenke, T., Leeb, H. & Schmiedmayer, J. Ramsey’s method of separated oscillating fields and its application to gravitationally induced quantum phase shifts. Phys. Rev. D 81, 065019 (2010).

    ADS  Article  Google Scholar 

  197. 197.

    Nesvizhevsky, V. V., Voronin, A. Y., Cubitt, R. & Protasov, K. V. Neutron whispering gallery. Nat. Phys. 6, 114–117 (2010).

    Article  Google Scholar 

  198. 198.

    Nesvizhevsky, V. V. & Voronin, A. Y. Centrifugal quantum states of neutrons. C. R. Phys. 12, 791–795 (2011).

    ADS  Article  Google Scholar 

  199. 199.

    Ignatovich, V. K. & Pokotilovski, Y. N. Limits on a nucleon–nucleon monopole–dipole coupling from spin relaxation of polarized ultra-cold neutrons in traps. Eur. Phys. J. C 64, 19–23 (2009).

    ADS  Article  Google Scholar 

  200. 200.

    Serebrov, A. P. New constraints for CP-violating forces between nucleons in the range 1 micrometer to 1 centimeter. Phys. Lett. B 680, 423–427 (2009).

    ADS  Article  Google Scholar 

  201. 201.

    Afach, S. et al. Constraining interactions mediated by axion-like particles with ultracold neutrons. Phys. Lett. B 745, 58–63 (2015).

    ADS  Article  Google Scholar 

  202. 202.

    Piegsa, F. M. & Pignol, G. Limits on the axial coupling constant of new light bosons. Phys. Rev. Lett. 108, 181801 (2012).

    ADS  Article  Google Scholar 

  203. 203.

    Mezei, F. Neutron spin echo: a new concept in polarized thermal neutron techniques. Z. Phys. 255, 146–160 (1972).

    ADS  Article  Google Scholar 

  204. 204.

    Frank, A. I. et al. New gravitational experiment with ultracold neutrons. JETP Lett. 86, 225–229 (2007).

    ADS  Article  Google Scholar 

  205. 205.

    Staudenmann, J. L., Werner, S. A., Colella, R. & Overhauser, A. W. Gravity and inertia in quantum mechanics. Phys. Rev. A 21, 1419–1438 (1980).

    ADS  Article  Google Scholar 

  206. 206.

    Dubbers, D. & Schmidt, M. G. The neutron and its role in cosmology and particle physics. Rev. Mod. Phys. 83, 1111–1171 (2011).

    ADS  Article  Google Scholar 

  207. 207.

    Dobrescu, B. A. & Mocioiu, I. Spin-dependent macroscopic forces from new particle exchange. J. High Energy Phys. 2006, 005 (2006).

    ADS  Article  Google Scholar 

  208. 208.

    Alighanbari, S., Giri, G. S., Constantin, F. L., Korobov, V. I. & Schiller, S. Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions. Nature 581, 152–158 (2020).

    ADS  Article  Google Scholar 

  209. 209.

    Biesheuvel, J. et al. Probing QED and fundamental constants through laser spectroscopy of vibrational transitions in HD+. Nat. Commun. 7, 10385 (2016).

    ADS  Article  Google Scholar 

  210. 210.

    Delaunay, C., Frugiuele, C., Fuchs, E. & Soreq, Y. Probing new spin-independent interactions through precision spectroscopy in atoms with few electrons. Phys. Rev. D 96, 115002 (2017).

    ADS  Article  Google Scholar 

  211. 211.

    Yan, H. & Snow, W. M. New limit on possible long-range parity-odd interactions of the neutron from neutron-spin rotation in liquid 4He. Phys. Rev. Lett. 110, 082003 (2013).

    ADS  Article  Google Scholar 

  212. 212.

    Yan, H. et al. Searching for new spin- and velocity-dependent interactions by spin relaxation of polarized 3He gas. Phys. Rev. Lett. 115, 182001 (2015).

    ADS  Article  Google Scholar 

  213. 213.

    Adelberger, E. G. & Wagner, T. A. Improved limits on long-range parity-odd interactions of the neutron. Phys. Rev. D 88, 031101 (2013).

    ADS  Article  Google Scholar 

  214. 214.

    Vasilakis, G., Brown, J. M., Kornack, T. W. & Romalis, M. V. Limits on new long range nuclear spin-dependent forces set with a K-3He comagnetometer. Phys. Rev. Lett. 103, 261801 (2009).

    ADS  Article  Google Scholar 

  215. 215.

    Haddock, C. et al. A search for possible long range spin dependent interactions of the neutron from exotic vector boson exchange. Phys. Lett. B 783, 227–233 (2018).

    ADS  Article  Google Scholar 

  216. 216.

    Fadeev, P. et al. Revisiting spin-dependent forces mediated by new bosons: potentials in the coordinate-space representation for macroscopic- and atomic-scale experiments. Phys. Rev. A 99, 022113 (2019).

    ADS  Article  Google Scholar 

  217. 217.

    Mantry, S., Pitschmann, M. & Ramsey-Musolf, M. J. Differences between Axions and Generic Light Scalars in Laboratory Experiments. (Verlag Deutsches Elektronen-Synchrotron, 2014).

    Google Scholar 

  218. 218.

    Safronova, M. S. et al. Search for new physics with atoms and molecules. Rev. Mod. Phys. 90, 025008 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  219. 219.

    Li, K. et al. Neutron limit on the strongly-coupled chameleon field. Phys. Rev. D 93, 062001 (2016).

    ADS  Article  Google Scholar 

  220. 220.

    Brax, P., van de Bruck, C., Davis, A., Khoury, J. & Weltman, A. Detecting dark energy in orbit: The cosmological chameleon. Phys. Rev. D 70, 123518 (2004).

    ADS  Article  Google Scholar 

  221. 221.

    Schmiedmayer, J. The equivalence of the gravitational and inertial mass of the neutron. Nucl. Instrum. Methods Phys. Res. A 284, 59–62 (1989).

    ADS  Article  Google Scholar 

  222. 222.

    Bergé, J. et al. MICROSCOPE mission: first constraints on the violation of the weak equivalence principle by a light scalar dilaton. Phys. Rev. Lett. 120, 141101 (2018).

    ADS  Article  Google Scholar 

  223. 223.

    Hentschel, K., Greenberger, D. & Weinert, F. Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy (Springer, 2009).

  224. 224.

    Aharonov, Y., Bergmann, P. G. & Lebowitz, J. L. Time symmetry in the quantum process of measurement. Phys. Rev. 134, B1410–B1416 (1964).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  225. 225.

    Reznik, B. & Aharonov, Y. Time-symmetric formulation of quantum mechanics. Phys. Rev. A 52, 2538–2550 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  226. 226.

    Bohm, D. A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166–179 (1952).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  227. 227.

    Snow, W. M. et al. Internal consistency of neutron coherent scattering length measurements from neutron interferometry and from neutron gravity reflectometry. Phys. Rev. D 101, 062004 (2020).

    ADS  Article  Google Scholar 

  228. 228.

    Tullney, K. et al. Constraints on spin-dependent short-range interaction between nucleons. Phys. Rev. Lett. 111, 100801 (2013).

    ADS  Article  Google Scholar 

  229. 229.

    Bulatowicz, M. et al. Laboratory search for a long-range T-odd, P-odd interaction from axionlike particles using dual-species nuclear magnetic resonance with polarized 129Xe and 131Xe gas. Phys. Rev. Lett. 111, 102001 (2013).

    ADS  Article  Google Scholar 

  230. 230.

    Jenke, T. et al. Gravity resonance spectroscopy and dark energy symmetron fields. Conference on Frontiers of Quantum and Mesoscopic Thermodynamics. Preprint at arXiv 2012.07472 (2020).

  231. 231.

    Voronin, V. V. et al. Eotvos-type experiment with cold neutron. Phys. Procedia 17, 232–238 (2011).

    ADS  Article  Google Scholar 

  232. 232.

    Voronin, V. V. et al. Analysis of spatial resolution of an experiment on verification of the equivalence principle for a neutron by the diffraction method. Tech. Phys. Lett. 43, 270–273 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  233. 233.

    Frank, A., Van Isacker, P. & Gómez-Camacho, J. Probing additional dimensions in the universe with neutron experiments. Phys. Lett. B 582, 15–20 (2004).

    ADS  Article  Google Scholar 

  234. 234.

    Angeli, I. & Marinova, K. P. Table of experimental nuclear ground state charge radii: An update. At. Data Nucl. Data Tables 99, 69–95 (2013).

    ADS  Article  Google Scholar 

  235. 235.

    Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. & Zeilinger, A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81, 5039–5043 (1998).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  236. 236.

    Rauch, D. et al. Cosmic Bell test using random measurement settings from high-redshift quasars. Phys. Rev. Lett. 121, 080403 (2018).

    ADS  Article  Google Scholar 

  237. 237.

    D’Ambrosio, V. et al. Experimental implementation of a Kochen-Specker set of quantum tests. Phys. Rev. X 3, 011012 (2013).

    Google Scholar 

  238. 238.

    Pan, J.-W., Bouwmeester, D., Daniell, M., Weinfurter, H. & Zeilinger, A. Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement. Nature 403, 515–519 (2000).

    ADS  MATH  Article  Google Scholar 

  239. 239.

    Erhard, M., Malik, M., Krenn, M. & Zeilinger, A. Experimental Greenberger–Horne–Zeilinger entanglement beyond qubits. Nat. Photonics 12, 759–764 (2018).

    ADS  Article  Google Scholar 

  240. 240.

    Cronenberg, G. et al. A gravity of Earth measurement with a qBOUNCE experiment. Proc. Sci. EPS-HEP2015, 408 (2016).

    Google Scholar 

  241. 241.

    Tan, W.-H. et al. New test of the gravitational inverse-square law at the submillimeter range with dual modulation and compensation. Phys. Rev. Lett. 116, 131101 (2016).

    ADS  Article  Google Scholar 

  242. 242.

    Kapner, D. J. et al. Tests of the gravitational inverse-square law below the dark-energy length scale. Phys. Rev. Lett. 98, 021101 (2007).

    ADS  Article  Google Scholar 

  243. 243.

    Geraci, A. A., Smullin, S. J., Weld, D. M., Chiaverini, J. & Kapitulnik, A. Improved constraints on non-Newtonian forces at 10 microns. Phys. Rev. D 78, 022002 (2008).

    ADS  Article  Google Scholar 

  244. 244.

    Chen, Y.-J. et al. Stronger limits on hypothetical Yukawa interactions in the 30–8000 nm range. Phys. Rev. Lett. 116, 221102 (2016).

    ADS  Article  Google Scholar 

  245. 245.

    Klimchitskaya, G. L. Recent breakthrough and outlook in constraining the non-Newtonian gravity and axion-like particles from Casimir physics. Eur. Phys. J. C 77, 315 (2017).

    ADS  Article  Google Scholar 

  246. 246.

    Baeßler, S., Nesvizhevsky, V. V., Protasov, K. V. & Voronin, A. Y. Constraint on the coupling of axionlike particles to matter via an ultracold neutron gravitational experiment. Phys. Rev. D 75, 075006 (2007).

    ADS  Article  Google Scholar 

  247. 247.

    Guigue, M., Jullien, D., Petukhov, A. K. & Pignol, G. Constraining short-range spin-dependent forces with polarized 3He. Phys. Rev. D 92, 114001 (2015).

    ADS  Article  Google Scholar 

  248. 248.

    Jaffe, M. et al. Testing sub-gravitational forces on atoms from a miniature in-vacuum source mass. Nat. Phys. 13, 938–942 (2017).

    Article  Google Scholar 

  249. 249.

    Upadhye, A. Dark energy fifth forces in torsion pendulum experiments. Phys. Rev. D 86, 102003 (2012).

    ADS  Article  Google Scholar 

  250. 250.

    Elder, B. et al. Classical symmetron force in Casimir experiments. Phys. Rev. D 101, 064065 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  251. 251.

    Upadhye, A. Symmetron dark energy in laboratory experiments. Phys. Rev. Lett. 110, 031301 (2013).

    ADS  Article  Google Scholar 

  252. 252.

    Sabulsky, D. O. et al. Experiment to detect dark energy forces using atom interferometry. Phys. Rev. Lett. 123, 061102 (2019).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The authors thank their co-workers and collaborators for their long-term efforts and support, and especially wish to thank the Institut Laue-Langevin (ILL), in Grenoble, France, for ongoing support and hospitality at instruments S18 and PF2. This work was supported by the Austrian Science Fund (FWF) project nos. P 30677, P 27666 and P 33279. Y.H. is partly supported by KAKENHI.

Author information

Affiliations

Authors

Contributions

Writing on quantum mechanics by S.S. and Y.H., dark interactions by R.I.P.S. and M.P., administration by Y.H. and H.A.

Corresponding authors

Correspondence to Stephan Sponar or Yuji Hasegawa.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sponar, S., Sedmik, R.I.P., Pitschmann, M. et al. Tests of fundamental quantum mechanics and dark interactions with low-energy neutrons. Nat Rev Phys 3, 309–327 (2021). https://doi.org/10.1038/s42254-021-00298-2

Download citation

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing