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.

Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond

Abstract

Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics.

Key points

  • The Aharonov–Bohm phase, acquired by charged particles encircling a confined magnetic flux, is topological, gauge invariant and realistic, highlighting the unique role of electromagnetic potentials in quantum mechanics.

  • The Aharonov–Bohm phase can be seen as a manifestation of Berry’s geometric phase accumulated whenever a quantum system is adiabatically transported around a cyclic circuit on an abstract surface in the parameter space (with additional generalizations to degenerate and open systems, and to non-adiabatic, non-cyclic, non-unitary evolutions).

  • The geometric phase is an example of a holonomy (failure of parallel transport around closed cycles to preserve the geometrical information being transported) and its profound role in physics.

  • The two main types of geometric phase in optics originate from ‘spin redirection’ (when light with a fixed state of polarization is changing direction) and from a slow change in polarization (of light propagating through an anisotropic medium in a fixed direction), giving rise to the Pancharatnam–Berry phase.

  • In condensed-matter physics, the geometric phase manifests itself in the electronic Bloch states, quantum Hall effect, electric polarization, exchange statistics and many other phenomena.

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: The Aharonov–Bohm effect.
Fig. 2: Parallel transport and holonomy within a vector bundle.
Fig. 3: Illustrations of the geometric phase in optics.
Fig. 4: Realizations of a spin-dependent geometric phase through a structured metasurface.
Fig. 5: Photonic quantum walk employing orbital angular momentum.
Fig. 6: Electronic Bloch states in the first Brillouin zone, and the quantum Hall effect.
Fig. 7: Exchange statistics in the case of anyons.

References

  1. 1.

    Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 11, 485 (1959).

    ADS  MathSciNet  MATH  Google Scholar 

  2. 2.

    Franz, W. Elektroneninterferenzen im Magnetfeld. Verh. Dtsch Phys. Ges. 20, 65 (1939).

    Google Scholar 

  3. 3.

    Ehrenberg, W. & Siday, R. E. The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. Lond. Sect. B 62, 8 (1949).

    ADS  MATH  Google Scholar 

  4. 4.

    Wu, T. T. & Yang, C. N. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845 (1975).

    ADS  MathSciNet  Google Scholar 

  5. 5.

    Peshkin, M. The Aharonov–Bohm effect: why it cannot be eliminated from quantum mechanics. Phys. Rep. 80, 375–386 (1981).

    ADS  MathSciNet  Google Scholar 

  6. 6.

    Olariu, S. & Popescu, I. I. The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339 (1985).

    ADS  Google Scholar 

  7. 7.

    Popescu, S. Dynamical quantum non-locality. Nat. Phys. 6, 151–153 (2010).

    Google Scholar 

  8. 8.

    Aharonov, Y., Cohen, E. & Rohrlich, D. Nonlocality of the Aharonov–Bohm effect. Phys. Rev. A 93, 042110 (2016).

    ADS  Google Scholar 

  9. 9.

    Vaidman, L. Role of potentials in the Aharonov–Bohm effect. Phys. Rev. A 86, 040101 (2012).

    ADS  Google Scholar 

  10. 10.

    Aharonov, Y., Cohen, E. & Rohrlich, D. Comment on “role of potentials in the Aharonov–Bohm effect”. Phys. Rev. A 92, 026101 (2015).

    ADS  Google Scholar 

  11. 11.

    Vaidman, L. Reply to “comment on ‘role of potentials in the Aharonov–Bohm effect’. Phys. Rev. A 92, 026102 (2015).

    ADS  Google Scholar 

  12. 12.

    Pearle, P. & Rizzi, A. Quantum-mechanical inclusion of the source in the Aharonov–Bohm effects. Phys. Rev. A 95, 052123 (2017).

    ADS  Google Scholar 

  13. 13.

    Pearle, P. & Rizzi, A. Quantized vector potential and alternative views of the magnetic Aharonov–Bohm phase shift. Phys. Rev. A 95, 052124 (2017).

    ADS  MathSciNet  Google Scholar 

  14. 14.

    Maudlin, T. Ontological clarity via canonical presentation: electromagnetism and the Aharonov–Bohm effect. Entropy 20, 465 (2018).

    ADS  Google Scholar 

  15. 15.

    Li, B., Hewak, D. W. & Wang, Q. J. The transition from quantum field theory to one-particle quantum mechanics and a proposed interpretation of Aharonov–Bohm effect. Found. Phys. 48, 837–852 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  16. 16.

    Weisskopf, V. F. in Lectures in Theoretical Physics Vol. 3 (ed. Britten, W. E.) 67 (Interscience, 1961).

  17. 17.

    Kretzschmar, M. Aharonov–Bohm scattering of a wave packet of finite extension. Z. Phys. 185, 84–96 (1965).

    ADS  MATH  Google Scholar 

  18. 18.

    Roy, S. M. Condition for nonexistence of Aharonov–Bohm effect. Phys. Rev. Lett. 44, 111–114 (1980).

    ADS  MathSciNet  MATH  Google Scholar 

  19. 19.

    Berry, M. in Fundamental Aspects of Quantum theory (eds Gorini V. & Frigerio, A.) 319–320 (Springer, 1986).

  20. 20.

    Aharonov, Y. & Casher, A. Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319–321 (1984).

    ADS  MathSciNet  Google Scholar 

  21. 21.

    Cimmino, A. et al. Observation of the topological Aharonov–Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63, 380–383 (1989).

    ADS  Google Scholar 

  22. 22.

    Elion, W. J., Wachters, J. J., Sohn, L. L. & Mooij, J. D. Observation of the Aharonov–Casher effect for vortices in Josephson-junction arrays. Phys. Rev. Lett. 71, 2311–2314 (1993).

    ADS  Google Scholar 

  23. 23.

    Koenig, M. et al. Direct observation of the Aharonov–Casher phase. Phys. Rev. Lett. 96, 076804 (2006).

    ADS  Google Scholar 

  24. 24.

    Chambers, R. G. Shift of an electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett. 5, 3–5 (1960).

    ADS  Google Scholar 

  25. 25.

    Tonomura, A. et al. Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett. 56, 792 (1986).

    ADS  Google Scholar 

  26. 26.

    Webb, R. A., Washburn, S., Umbach, C. P. & Laibowitz, R. B. Observation of h/e Aharonov–Bohm oscillations in normal-metal rings. Phys. Rev. Lett. 54, 2696–2699 (1985).

    ADS  Google Scholar 

  27. 27.

    Yacoby, Y., Heiblum, M., Mahalu, D. & Shtrikman, H. Coherence and phase sensitive measurements in a quantum dot. Phys. Rev. Lett. 74, 4047–4050 (1995).

    ADS  Google Scholar 

  28. 28.

    Bachtold, A. et al. Aharonov–Bohm oscillations in carbon nanotubes. Nature 397, 673–675 (1999).

    ADS  Google Scholar 

  29. 29.

    Ji, Y. et al. An electronic Mach–Zehnder interferometer. Nature 422, 415–418 (2003).

    ADS  Google Scholar 

  30. 30.

    Peng, H. et al. Aharonov–Bohm interference in topological insulator nanoribbons. Nat. Mater. 9, 225–229 (2010).

    ADS  Google Scholar 

  31. 31.

    Bardarson, J. H., Brouwer, P. W. & Moore, J. E. Aharonov–Bohm oscillations in disordered topological insulator nanowires. Phys. Rev. Lett. 105, 156803 (2010).

    ADS  Google Scholar 

  32. 32.

    Zhang, Y. & Vishwanath, A. Anomalous Aharonov–Bohm conductance oscillations from topological insulator surface states. Phys. Rev. Lett. 105, 206601 (2010).

    ADS  Google Scholar 

  33. 33.

    Aidelsburger, M. et al. Experimental realization of strong effective magnetic fields in an optical lattice. Phys. Rev. Lett. 107, 255301 (2011).

    ADS  Google Scholar 

  34. 34.

    Duca, L. et al. An Aharonov–Bohm interferometer for determining Bloch band topology. Science 347, 288–292 (2015).

    ADS  Google Scholar 

  35. 35.

    Noguchi, A., Shikano, Y., Toyoda, K. & Urabe, S. Aharonov–Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap. Nat. Commun. 5, 3868 (2014).

    ADS  Google Scholar 

  36. 36.

    Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  37. 37.

    Born, M. & Fock, V. A. Beweis des Adiabatensatzes. Z. Phys. A 51, 165–180 (1927).

    MATH  Google Scholar 

  38. 38.

    Simon, B. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167 (1983).

    ADS  MathSciNet  Google Scholar 

  39. 39.

    Wilczek, F. & Shapere, A. Geometric Phases in Physics. (World Scientific:, 1989).

  40. 40.

    Markovski, B. & Vinitsky, S. I. Topological Phases in Quantum Theory. (World Scientific:, 1989).

  41. 41.

    Zwanziger, J. W., Koenig, M. & Pines, A. Berry’s phase. Annu. Rev. Phys. Chem. 41, 601–646 (1990).

    ADS  Google Scholar 

  42. 42.

    Anandan, J. The geometric phase. Nature 360, 307–313 (1992).

    ADS  Google Scholar 

  43. 43.

    Li, H.-Z Global Properties of Simple Quantum Systems — Berry’s Phase and Others. (Shanghai Scientific and Technical Publishers:, 1998).

  44. 44.

    Bohm, A, Mostafazadeh, A, Koizumi, H, Niu, Q. & Zwanziger, J. The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics. (Springer Verlag:, 2003).

  45. 45.

    Chruscinski, D. & Jamiolkowski, A. Geometric Phases in Classical and Quantum Mechanics. (Birkhäuser:, 2004).

  46. 46.

    Berry, M. V. Geometric phases. WordPress https://michaelberryphysics.files.wordpress.com/2018/03/berryd.pdf (2018).

  47. 47.

    Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010).

    ADS  MathSciNet  MATH  Google Scholar 

  48. 48.

    Berry, M. Anticipations of the geometric phase. Phys. Today 43, 34–40 (1990).

    ADS  Google Scholar 

  49. 49.

    Berry, M. Geometric phase memories. Nat. Phys. 6, 148–150 (2010).

    Google Scholar 

  50. 50.

    Pancharatnam, S. Generalized theory of interference and its applications. Proc. Indian Acad. Sci. A 44, 247–262 (1956).

    MathSciNet  Google Scholar 

  51. 51.

    Berry, M. V. The adiabatic phase and Pancharatnam’s phase for polarized light. J. Mod. Opt. 34, 1401–1407 (1987).

    ADS  MathSciNet  MATH  Google Scholar 

  52. 52.

    Longuet-Higgins, H. C., Öpik, U., Pryce, M. H. L. & Sack, R. A. Studies of the Jahn–Teller effect. II. The dynamical problem. Proc. R. Soc. Lond. A 244, 1–16 (1958).

    ADS  MATH  Google Scholar 

  53. 53.

    Mead, C. A. & Truhlar, D. G. On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. J. Chem. Phys. 70, 2284–2296 (1979).

    ADS  Google Scholar 

  54. 54.

    Hamilton, W. R. Third supplement to an essay on the theory of systems of rays. Trans. R. Ir. Acad. 17, 1–144 (1837).

    Google Scholar 

  55. 55.

    Lloyd, H. On the phenomena presented by light in its passage along the axes of biaxial crystals. Phil. Mag. 1, 112–120; 207–210 (1833).

  56. 56.

    Bortolotti, E. Memories and notes presented by fellows. Rend. R. Acc. Naz. Linc. 4, 552 (1926).

    Google Scholar 

  57. 57.

    Rytov, S. M. On transition from wave to geometrical optics. Dokl. Akad. Nauk SSSR 18, 263–266 (1938).

    MATH  Google Scholar 

  58. 58.

    Vladimirskii, V. V. The rotation of a polarization plane for curved light ray. Dokl. Akad. Nauk. SSSR 31, 222–224 (1941).

    Google Scholar 

  59. 59.

    Budden, K. G. & Smith, M. S. Phase memory and additional memory in WKB solutions for wave propagation in stratified media. Proc. R. Soc. Lond. A 350, 27–46 (1976).

    ADS  MATH  Google Scholar 

  60. 60.

    Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111 (1984).

    ADS  MathSciNet  Google Scholar 

  61. 61.

    Aharonov, Y. & Anandan, J. Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593 (1987).

    ADS  MathSciNet  Google Scholar 

  62. 62.

    Berry, M. V. Quantum phase corrections from adiabatic iteration. Proc. R. Soc. Lond. A 414, 31–46 (1987).

    ADS  MathSciNet  Google Scholar 

  63. 63.

    Samuel, J. & Bhandari, R. General setting for Berry’s phase. Phys. Rev. Lett. 60, 2339 (1988).

    ADS  MathSciNet  Google Scholar 

  64. 64.

    Hannay, J. H. Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A 18, 221–230 (1985).

    ADS  MathSciNet  Google Scholar 

  65. 65.

    Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747 (1989).

    ADS  Google Scholar 

  66. 66.

    Meir, Y., Gefen, Y. & Entin-Wohlman, O. Universal effects of spin-orbit scattering in mesoscopic systems. Phys. Rev. Lett. 63, 798 (1989).

    ADS  Google Scholar 

  67. 67.

    Loss, D., Goldbart, P. & Balatsky, A. V. Berry’s phase and persistent charge and spin currents in textured mesoscopic rings. Phys. Rev. Lett. 65, 1655 (1990).

    ADS  Google Scholar 

  68. 68.

    Nagasawa, F., Frustaglia, D., Saarikoski, H., Richter, K. & Nitta, J. Control of the spin geometric phase in semiconductor quantum rings. Nat. Commun. 4, 2526 (2013).

    ADS  Google Scholar 

  69. 69.

    Garrison, J. C. & Wright, E. M. Complex geometrical phases for dissipative systems. Phys. Lett. A 128, 177–181 (1988).

    ADS  MathSciNet  Google Scholar 

  70. 70.

    Berry, M. V. Geometric amplitude factors in adiabatic quantum transitions. Proc. R. Soc. Lond. A 430, 405–411 (1990).

    ADS  MathSciNet  Google Scholar 

  71. 71.

    Zwanziger, J. W., Rucker, S. P. & Chingas, G. C. Measuring the geometric component of the transition probability in a two-level system. Phys. Rev. A 43, 3232–3240 (1991).

    ADS  Google Scholar 

  72. 72.

    Kepler, T. B. & Kagan, M. L. Geometric phase shifts under adiabatic parameter changes in classical dissipative systems. Phys. Rev. Lett. 66, 847 (1991).

    ADS  Google Scholar 

  73. 73.

    Ning, C. Z. & Haken, H. Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors. Phys. Rev. Lett. 68, 2109–2122 (1992).

    ADS  MathSciNet  MATH  Google Scholar 

  74. 74.

    Bliokh, K. Y. The appearance of a geometric-type instability in dynamic systems with adiabatically varying parameters. J. Phys. A 32, 2551 (1999).

    ADS  MathSciNet  MATH  Google Scholar 

  75. 75.

    Carollo, A., Fuentes-Guridi, I., Santos, M. F. & Vedral, V. Geometric phase in open systems. Phys. Rev. Lett. 90, 160402 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  76. 76.

    Dietz, B. et al. Exceptional points in a microwave billiard with time-reversal invariance violation. Phys. Rev. Lett. 106, 150403 (2011).

    ADS  Google Scholar 

  77. 77.

    Whitney, R. S. & Gefen, Y. Berry phase in a nonisolated system. Phys. Rev. Lett. 90, 190402 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  78. 78.

    Whitney, R. S., Makhlin, Y., Shnirman, A. & Gefen, Y. Geometric nature of the environment-induced Berry phase and geometric dephasing. Phys. Rev. Lett. 94, 070407 (2005).

    ADS  MathSciNet  Google Scholar 

  79. 79.

    Berger, S. et al. Measurement of geometric dephasing using a superconducting qubit. Nat. Commun. 6, 8757 (2015).

    ADS  Google Scholar 

  80. 80.

    Gaitan, F. Berry’s phase in the presence of a stochastically evolving environment: a geometric mechanism for energy-level broadening. Phys. Rev. A 58, 1665 (1998).

    ADS  MathSciNet  Google Scholar 

  81. 81.

    De Chiara, G. & Palma, G. M. Berry phase for a spin 1/2 particle in a classical fluctuating field. Phys. Rev. Lett. 91, 90404 (2003).

    Google Scholar 

  82. 82.

    Dembowski, C. et al. Encircling an exceptional point. Phys. Rev. E 69, 056216 (2004).

    ADS  Google Scholar 

  83. 83.

    Mailybaev, A. A., Kirillov, O. N. & Seyranian, A. P. Geometric phase around exceptional points. Phys. Rev. A 72, 014104 (2005).

    ADS  Google Scholar 

  84. 84.

    Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015).

    ADS  Google Scholar 

  85. 85.

    Zygelman, B. Appearance of gauge potentials in atomic collision physics. Phys. Lett. A 125, 476–481 (1987).

    ADS  Google Scholar 

  86. 86.

    Holstein, B. R. The Aharonov–Bohm effect and variations. Contemp. Phys. 36, 93–102 (1995).

    ADS  Google Scholar 

  87. 87.

    Aharonov, Y. et al. Aharonov–Bohm and Berry phases for a quantum cloud of charge. Phys. Rev. Lett. 73, 918 (1994).

    ADS  Google Scholar 

  88. 88.

    Bott, R. & Chern, S. S. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, 71–112 (1965).

    MathSciNet  MATH  Google Scholar 

  89. 89.

    Chern, S. S. Characteristic classes of Hermitian manifolds. Ann. Math. 47, 85–121 (1946).

    MathSciNet  MATH  Google Scholar 

  90. 90.

    Dirac, P. A. M. Quantised singularities in the electromagnetic field. Proc. R. Soc. A 133, 60–72 (1931).

    ADS  MATH  Google Scholar 

  91. 91.

    Pietilä, V. & Möttönen, M. Creation of Dirac monopoles in spinor Bose–Einstein condensates. Phys. Rev. Lett. 103, 030401 (2009).

    ADS  Google Scholar 

  92. 92.

    Ray, M. W., Ruokokoski, E., Kandel, S., Möttönen, M. & Hall, D. S. Observation of Dirac monopoles in a synthetic magnetic field. Nature 505, 657–660 (2014).

    ADS  Google Scholar 

  93. 93.

    Bernevig, B. A. It’s been a Weyl coming. Nat. Phys. 11, 698–699 (2015).

    Google Scholar 

  94. 94.

    Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  95. 95.

    Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724–727 (2015).

    Google Scholar 

  96. 96.

    Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nat. Phys. 11, 728–732 (2015).

    Google Scholar 

  97. 97.

    Xu, S.-Y. et al. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nat. Phys. 11, 748–754 (2015).

    Google Scholar 

  98. 98.

    Lu, L., Fu, L., Joannopoulos, J. D. & Soljaĉić, M. Weyl points and line nodes in gyroid photonic crystals. Nat. Photonics 7, 294–299 (2013).

    ADS  Google Scholar 

  99. 99.

    Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  100. 100.

    Bliokh, K. Y. Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium. J. Opt. A 11, 094009 (2009).

    ADS  Google Scholar 

  101. 101.

    Ross, J. N. The rotation of the polarization in low birefringence monomode optical fibres due to geometric effects. Opt. Quantum Electron. 16, 455–461 (1984).

    ADS  Google Scholar 

  102. 102.

    Tomita, A. & Chiao, R. Y. Observation of Berry’s topological phase by use of an optical fiber. Phys. Rev. Lett. 57, 937 (1986).

    ADS  Google Scholar 

  103. 103.

    Chiao, R. Y. & Wu, Y. S. Manifestations of Berry’s topological phase for the photon. Phys. Rev. Lett. 57, 933 (1986).

    ADS  Google Scholar 

  104. 104.

    Mathur, H. Thomas precession, spin-orbit interaction, and Berry’s phase. Phys. Rev. Lett. 67, 3325 (1991).

    ADS  MathSciNet  MATH  Google Scholar 

  105. 105.

    Berard, A. & Mohrbach, H. Spin Hall effect and Berry phase of spinning particles. Phys. Lett. A 352, 190–195 (2006).

    ADS  MathSciNet  MATH  Google Scholar 

  106. 106.

    Bliokh, K. Y., Alonso, M. A., Ostrovskaya, E. A. & Aiello, A. Angular momenta and spin-orbit interaction of nonparaxial light in free space. Phys. Rev. A 82, 063825 (2010).

    ADS  Google Scholar 

  107. 107.

    Jones, R. C. A new calculus for the treatment of optical systems I. Description and discussion of the calculus. J. Opt. Soc. Am. 31, 488–493 (1941).

    ADS  MATH  Google Scholar 

  108. 108.

    Simon, R., Kimble, H. J. & Sudarshan, E. C. G. Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment. Phys. Rev. Lett. 61, 19 (1988).

    ADS  Google Scholar 

  109. 109.

    Bhandari, R. Polarization of light and topological phases. Phys. Rep. 281, 1–64 (1997).

    ADS  Google Scholar 

  110. 110.

    Berry, M. V. & Klein, S. Geometric phases from stacks of crystal plates. J. Mod. Opt. 43, 165–180 (1996).

    ADS  MathSciNet  MATH  Google Scholar 

  111. 111.

    Bomzon, Z., Biener, G., Kleiner, V. & Hasman, E. Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings. Opt. Lett. 27, 1141–1143 (2002).

    ADS  Google Scholar 

  112. 112.

    Biener, G., Niv, A., Kleiner, V. & Hasman, E. Formation of helical beams by use of Pancharatnam–Berry phase optical elements. Opt. Lett. 27, 1875–1877 (2002).

    ADS  Google Scholar 

  113. 113.

    Marrucci, L., Manzo, C. & Paparo, D. Pancharatnam–Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation. Appl. Phys. Lett. 88, 221102 (2006).

    ADS  Google Scholar 

  114. 114.

    Lin, D., Fan, P., Hasman, E. & Brongersma, M. L. Dielectric gradient metasurface optical elements. Science 345, 298–302 (2014).

    ADS  Google Scholar 

  115. 115.

    Hasman, E., Kleiner, V., Biener, G. & Niv, A. Polarization dependent focusing lens by use of quantized Phancharatnam–Berry phase diffractive optics. Appl. Phys. Lett. 82, 328 (2003).

    ADS  Google Scholar 

  116. 116.

    Mansuripur, M., Zakharian, A. R. & Wright, E. M. Spin and orbital angular momenta of light refrected from a cone. Phys. Rev. A 81, 033813 (2011).

    ADS  Google Scholar 

  117. 117.

    Bouchard, F., Mand, H., Mirhosseini, M., Karimi, E. & Boyd, R. W. Achromatic orbital angular momentum generator. New J. Phys. 16, 123006 (2014).

    ADS  Google Scholar 

  118. 118.

    Radwell, N., Hawley, R. D., Gotte, J. B. & Franke-Arnold, S. Achromatic vector vortex beams from a glass cone. Nat. Commun. 7, 10564 (2016).

    ADS  Google Scholar 

  119. 119.

    Devlin, R. C., Ambrosio, A., Rubin, N. A., Mueller, J. P. B. & Capasso, F. Arbitrary spin-to-orbital angular momentum conversion of light. Science 358, 896–901 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  120. 120.

    Padgett, M. J. & Courtial, J. Poincaré-sphere equivalent for light beams containing orbital angular momentum. Opt. Lett. 24, 430–432 (1999).

    ADS  Google Scholar 

  121. 121.

    Galvez, E. J. et al. Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum. Phys. Rev. Lett. 90, 203901 (2003).

    ADS  Google Scholar 

  122. 122.

    Calvo, G. F. Wigner representation and geometric transformations of optical orbital angular momentum spatial modes. Opt. Lett. 30, 1207–1209 (2005).

    ADS  Google Scholar 

  123. 123.

    Karimi, E., Slussarenko, S., Piccirillo, B., Marrucci, L. & Santamato, E. Polarization-controlled evolution of light transverse modes and associated Pancharatnam geometric phase in orbital angular momentum. Phys. Rev. A 81, 053813 (2010).

    ADS  Google Scholar 

  124. 124.

    Milione, G., Sztul, H. I., Nolan, D. A. & Alfano, R. R. Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light. Phys. Rev. Lett. 107, 053601 (2011).

    ADS  Google Scholar 

  125. 125.

    Malhotra, T. et al. Measuring geometric phase without interferometry. Phys. Rev. Lett. 120, 233602 (2018).

    ADS  Google Scholar 

  126. 126.

    Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics 3, 161–204 (2011).

    ADS  Google Scholar 

  127. 127.

    Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185 (1992).

    ADS  Google Scholar 

  128. 128.

    Beijersbergen, M. W., Coerwinkel, R. P. C., Kristensen, M. & Woerdman, J. P. Helical-wavefront laser beams produced with a spiral phaseplate. Opt. Commun. 112, 321–327 (1994).

    ADS  Google Scholar 

  129. 129.

    Heckenberg, N. R., McDuff, R., Smith, C. P., Rubinsztein-Dunlop, H. & Wegener, M. J. Laser beams with phase singularities. Opt. Quantum Electron. 24, S951–S962 (1992).

    Google Scholar 

  130. 130.

    Bazhenov, V. Yu, Soskin, M. S. & Vasnetsov, M. V. Screw dislocations in light wavefronts. J. Mod. Opt. 39, 985–990 (1992).

    ADS  Google Scholar 

  131. 131.

    Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 96, 163905 (2006).

    ADS  Google Scholar 

  132. 132.

    Liberman, V. S. & Zeldovich, B. Y. Spin-orbit interaction of a photon in an inhomogeneous medium. Phys. Rev. A 46, 5199–5520 (1992).

    ADS  Google Scholar 

  133. 133.

    Bliokh, K. Y. et al. Spin-orbit interactions of light. Nat. Photonics 9, 796–808 (2015).

    ADS  Google Scholar 

  134. 134.

    Brasselet, E., Murazawa, N., Misawa, H. & Juodkazis, S. Optical vortices from liquid crystal droplets. Phys. Rev. Lett. 103, 103903 (2009).

    ADS  Google Scholar 

  135. 135.

    Larocque, H. et al. Arbitrary optical wavefront shaping via spin-to-orbit coupling. J. Opt. 18, 124002 (2016).

    ADS  Google Scholar 

  136. 136.

    Quabis, S., Dorn, R. & Leuchs, G. Generation of a radially polarized doughnut mode of high quality. Appl. Phys. B 81, 597–600 (2005).

    ADS  Google Scholar 

  137. 137.

    Cardano, F. et al. Polarization pattern of vector vortex beams generated by q-plates with different topological charges. Appl. Opt. 51, C1–C6 (2012).

    Google Scholar 

  138. 138.

    Beresna, M., Gecevicius, M., Kazansky, P. G. & Gertus, T. Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass. Appl. Phys. Lett. 98, 201101 (2011).

    ADS  Google Scholar 

  139. 139.

    Cardano, F., Karimi, E., Marrucci, L., de Lisio, C. & Santamato, E. Generation and dynamics of optical beams with polarization singularities. Opt. Express 21, 8815–8820 (2013).

    ADS  Google Scholar 

  140. 140.

    Bauer, T. et al. Observation of optical polarization Möbius strips. Science 347, 964–966 (2015).

    ADS  Google Scholar 

  141. 141.

    Karimi, E. et al. Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface. Light Sci. Appl. 3, e167 (2014).

    MathSciNet  Google Scholar 

  142. 142.

    Bouchard, F. et al. Optical spin-to-orbital angular momentum conversion in ultra-thin metasurfaces with arbitrary topological charges. Appl. Phys. Lett. 105, 101905 (2014).

    ADS  Google Scholar 

  143. 143.

    Bliokh, K. Y., Niv, A., Kleiner, V. & Hasman, E. Geometrodynamics of spinning light. Nat. Photonics 2, 748–753 (2008).

    ADS  Google Scholar 

  144. 144.

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

    ADS  Google Scholar 

  145. 145.

    Merano, M., Aiello, A., Van Exter, M. P. & Woerdman, J. P. Observing angular deviations in the specular reflection of a light beam. Nat. Photonics 3, 337–340 (2009).

    ADS  Google Scholar 

  146. 146.

    Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  147. 147.

    Cardano, F. et al. Quantum walks and wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 1, e1500087 (2015).

    ADS  Google Scholar 

  148. 148.

    Cardano, F. et al. Statistical moments of quantum-walk dynamics reveal topological quantum transitions. Nat. Commun. 7, 11439 (2016).

    ADS  Google Scholar 

  149. 149.

    Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Commun. 8, 15516 (2017).

    ADS  Google Scholar 

  150. 150.

    Sephton, B. et al. A versatile quantum walk resonator with bright classical light. Preprint at arXiv https://arxiv.org/abs/1810.06850 (2018).

  151. 151.

    Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators. Nat. Phys. 8, 285–291 (2012).

    Google Scholar 

  152. 152.

    Childs, A. M. Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009).

    ADS  MathSciNet  Google Scholar 

  153. 153.

    Bliokh, K. Y., Bliokh, Y. P., Savel’Ev, S. & Nori, F. Semiclassical dynamics of electron wave packet states with phase vortices. Phys. Rev. Lett. 99, 190404 (2007).

    ADS  Google Scholar 

  154. 154.

    Karimi, E., Marrucci, L., Grillo, V. & Santamato, E. Spin-to-orbital angular momentum conversion and spin-polarization filtering in electron beams. Phys. Rev. Lett. 108, 044801 (2012).

    ADS  Google Scholar 

  155. 155.

    Sundaram, G. & Niu, Q. Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915 (1999).

    ADS  Google Scholar 

  156. 156.

    Bliokh, K. Y. & Bliokh, Y. P. Spin gauge fields: from Berry phase to topological spin transport and Hall effects. Ann. Phys. 319, 13 (2005).

    ADS  MathSciNet  MATH  Google Scholar 

  157. 157.

    Wyckoff, R. W. G The Analytical Expression of the Results of the Theory of Space Groups(Carnegie Institution of Washington:, 1922).

  158. 158.

    Thouless, J. D., Kohmoto, M., Nightingale, P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

    ADS  Google Scholar 

  159. 159.

    Jungwirth, T., Niu, Q. & MacDonald, A. H. Anomalous Hall effect in ferromagnetic semiconductors. Phys. Rev. Lett. 88, 207208 (2002).

    ADS  Google Scholar 

  160. 160.

    Karplus, R. & Luttinger, J. M. Hall effect in ferromagnetics. Phys. Rev. 95, 1154 (1954).

    ADS  MATH  Google Scholar 

  161. 161.

    Smit, J. The spontaneous Hall effect in ferromagnetics I. Physica 21, 877 (1955).

    ADS  Google Scholar 

  162. 162.

    Resta, R. Theory of the electric polarization in crystals. Ferroelectrics 136, 51–55 (1992).

    Google Scholar 

  163. 163.

    King-Smith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651 (1993).

    ADS  Google Scholar 

  164. 164.

    Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864 (1964).

    ADS  MathSciNet  Google Scholar 

  165. 165.

    Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965).

    ADS  MathSciNet  Google Scholar 

  166. 166.

    Resta, R. & Vanderbilt, D. in Physics of Ferroelectrics Topics in Applied Physics Vol. 105 (eds Ascheron, C. E. & Skolaut, W.) 31–68 (Springer-Verlag Berlin, 2007).

  167. 167.

    Berry, M. V. & Robbins, J. M. Indistinguishability for quantum particles: spin, statistics and the geometric phase. Proc. R. Soc. A 453, 1771–1790 (1997).

    ADS  MathSciNet  MATH  Google Scholar 

  168. 168.

    Berry, M. V. & Robbins, J. M. Quantum Indistinguishability: alternative constructions of the transported basis. J. Phys. A 33, L207–L214 (2000).

    ADS  MathSciNet  MATH  Google Scholar 

  169. 169.

    Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Nuovo Cim. 37, 1–23 (1977).

    ADS  Google Scholar 

  170. 170.

    Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957 (1982).

    ADS  MathSciNet  Google Scholar 

  171. 171.

    Campagnano, G. et al. Hanbury Brown–Twiss interference of anyons. Phys. Rev. Lett. 109, 106802 (2012).

    ADS  Google Scholar 

  172. 172.

    Altland, A., Gefen, Y. & Rosenow, B. Intermediate fixed point in a Luttinger liquid with elastic and dissipative backscattering. Phys. Rev. B 92, 085124 (2015).

    ADS  Google Scholar 

  173. 173.

    Law, K. T., Feldman, D. E. & Gefen, Y. Electronic Mach-Zehnder interferometer as a tool to probe fractional statistics. Phys. Rev. B 74, 045319 (2006).

    ADS  Google Scholar 

  174. 174.

    Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722 (1984).

    ADS  Google Scholar 

  175. 175.

    Thouless, D. J. & Gefen, Y. Fractional quantum Hall effect and multiple Aharonov–Bohm periods. Phys. Rev. Lett. 66, 806 (1991).

    ADS  Google Scholar 

  176. 176.

    Chamon, C. C., Freed, D. & Wen, X. G. Tunneling and quantum noise in one-dimensional Luttinger liquids. Phys. Rev. B 51, 2363 (1995).

    ADS  Google Scholar 

  177. 177.

    Feldman, D. E., Gefen, Y., Kitaev, A., Law, K. T. & Stern, A. Shot noise in an anyonic Mach–Zehnder interferometer. Phys. Rev. B 76, 085333 (2007).

    ADS  Google Scholar 

  178. 178.

    Stern, A. Non-Abelian states of matter. Nature 464, 187–193 (2010).

    ADS  Google Scholar 

  179. 179.

    Wilson, K. G. Confinement of quarks. Phys. Rev. D 10, 2445 (1974).

    ADS  Google Scholar 

  180. 180.

    Sonoda, H. Berry’s phase in chiral gauge theories. Nucl. Phys. B 266, 410–422 (1986).

    ADS  MathSciNet  Google Scholar 

  181. 181.

    Aitchison, I. J. R. & Hey, A. J. G. Gauge Theories in Particle Physics: A Practical Introduction. Volume 2: Non-Abelian Gauge Theories: QCD and the Electroweak Theory(CRC Press:, 2012).

  182. 182.

    Dowker, J. S. A gravitational Aharonov–Bohm effect. Nuovo Ciemento B 52, 129–135 (1967).

    ADS  Google Scholar 

  183. 183.

    Ford, L. H. & Vilenkin, A. A gravitational analogue of the Aharonov–Bohm effect. J. Phys. A 14, 2353 (1981).

    ADS  MathSciNet  Google Scholar 

  184. 184.

    Datta, D. P. Geometric phase in vacuum instability: applications in quantum cosmology. Phys. Rev. D 48, 5746 (1993).

    ADS  MathSciNet  Google Scholar 

  185. 185.

    Berry, M. V. et al. Wavefront dislocations in the Aharonov–Bohm effect and its water wave analogue. Eur. J. Phys. 1, 154 (1980).

    MathSciNet  Google Scholar 

  186. 186.

    Son, D. T. & Yamamoto, N. Berry curvature, triangle anomalies, and the chiral magnetic effect in Fermi liquids. Phys. Rev. Lett. 109, 81602 (2012).

    ADS  Google Scholar 

  187. 187.

    Mead, C. A. The geometric phase in molecular systems. Rev. Mod. Phys. 64, 51 (1992).

    ADS  MathSciNet  Google Scholar 

  188. 188.

    Kuppermann, A. & Wu, Y. S. M. The geometric phase effect shows up in chemical reactions. Chem. Phys. Lett. 205, 577–586 (1993).

    ADS  Google Scholar 

  189. 189.

    Kendrick, B. K., Hazra, J. & Balakrishnan, N. The geometric phase controls ultracold chemistry. Nat. Commun. 6, 7918 (2015).

    ADS  Google Scholar 

  190. 190.

    Berry, M. V. & Shukla, P. Geometric phase curvature for random states. J. Phys. A 51, 475101 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  191. 191.

    Zanardi, P. & Rasetti, M. Holonomic quantum computation. Phys. Lett. A 264, 94–99 (1999).

    ADS  MathSciNet  MATH  Google Scholar 

  192. 192.

    Jones, J. A., Vedral, V., Ekert, A. & Castagnoli, G. Geometric quantum computation using nuclear magnetic resonance. Nature 403, 869–871 (2000).

    ADS  Google Scholar 

  193. 193.

    Duan, L. M., Cirac, J. I. & Zoller, P. Geometric manipulation of trapped ions for quantum computation. Science 292, 1695–1697 (2001).

    ADS  Google Scholar 

  194. 194.

    Vedral, V. Geometric phases and topological quantum computation. Int. J. Quant. Inf. 1, 1–23 (2003).

    MATH  Google Scholar 

  195. 195.

    Leek, P. J. et al. Observation of Berry’s phase in a solid-state qubit. Science 318, 1889–1892 (2007).

    ADS  MathSciNet  MATH  Google Scholar 

  196. 196.

    Nayak, C. et al. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  197. 197.

    Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M. P. Non-Abelian statistics and topological quantum information processing in 1D wire networks. Nat. Phys. 7, 412–417 (2011).

    Google Scholar 

  198. 198.

    Berger, S. et al. Measurement of geometric dephasing using a superconducting qubit. Nat. Commun. 6, 8757 (2015).

    ADS  Google Scholar 

  199. 199.

    Yale, C. G. et al. Optical manipulation of the Berry phase in a solid-state spin qubit. Nat. Photon. 10, 184–189 (2016).

    ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by Canada Research Chair (CRC), Canada Foundation for Innovation (CFI), Canada First Excellence Research Fund (CFREF) Program, DFG grants no. MI 658/10-1, no. RO 2247/8-1 and CRC 183, Leverhulme Trust and the Italia-Israel project QUANTRA.

Author information

Affiliations

Authors

Contributions

All authors have read, discussed and contributed to the writing of the manuscript.

Corresponding authors

Correspondence to Eliahu Cohen or Ebrahim Karimi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

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

Glossary

Topological invariants

Any properties of a topological space that are invariant under homeomorphisms.

Winding number

The number of times that a closed curve travels anticlockwise around a point on a surface.

Connection

Operation that allows parallel transport over the vector bundle.

Vector bundle

A family of vector spaces.

Chern class

A certain topological invariant associated with vector bundles on smooth manifolds.

Jones vectors

The 2D vectors describing polarization of light.

Quantum walk

The quantum analogue of classical random walk taking advantage of quantum coherent superposition.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cohen, E., Larocque, H., Bouchard, F. et al. Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nat Rev Phys 1, 437–449 (2019). https://doi.org/10.1038/s42254-019-0071-1

Download citation

Further reading

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