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  • Letter
  • Open Access

Majorana stellar representation of twisted photons

Nicolas Fabre1,2, Andrei B. Klimov3, Romain Murenzi4, Jean-Pierre Gazeau5, and Luis L. Sánchez-Soto1,6

  • 1Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
  • 2Telecom Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France
  • 3Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
  • 4The World Academy of Sciences, TWAS, ICTP, Strada Costiera 1, 34151 Trieste, Italy
  • 5Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
  • 6Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany

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Phys. Rev. Research 5, L032006 – Published 12 July, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L032006

Abstract

Majorana stellar representation, which visualizes a quantum spin as points on the Bloch sphere, allows quantum mechanics to accommodate the concept of trajectory, the hallmark of classical physics. We extend this notion to the discrete cylinder, which is the phase space of the canonical pair angle and orbital angular momentum. We demonstrate that the geometrical properties of the ensuing constellations aptly encapsulate the quantumness of the state.

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References (90)

  1. E. Majorana, Atomi orientati in campo magnetico variabile, Nuovo Cimento 9, 43 (1932).
  2. H. Mäkelä and A. Messina, n-qubit states as points on the Bloch sphere, Phys. Scr. 2010, 014054 (2010).
  3. A. R. U. Devi, Sudha, and A. K. Rajagopal, Majorana representation of symmetric multiqubit states, Quantum Inf. Process. 11, 685 (2012).
  4. C. Yang, H. Guo, L.-B. Fu, and S. Chen, Characterization of symmetry-protected topological phases in polymerized models by trajectories of Majorana stars, Phys. Rev. B 91, 125132 (2015).
  5. Y.-G. Su, F. Yao, H.-B. Liang, Y.-M. Che, L.-B. Fu, and X.-G. Wang, Majorana stellar representation for mixed-spin (s,1/2) systems, Chin. Phys. B 30, 030303 (2021).
  6. P. Ribeiro, J. Vidal, and R. Mosseri, Thermodynamical Limit of the Lipkin-Meshkov-Glick Model, Phys. Rev. Lett. 99, 050402 (2007).
  7. P. Ribeiro and R. Mosseri, Entanglement in the Symmetric Sector of n Qubits, Phys. Rev. Lett. 106, 180502 (2011).
  8. H. D. Liu and L. B. Fu, Berry phase and quantum entanglement in Majorana's stellar representation, Phys. Rev. A 94, 022123 (2016).
  9. A. Lamacraft, Low-energy dynamics of spinor condensates, Phys. Rev. B 81, 184526 (2010).
  10. B. Lian, T.-L. Ho, and H. Zhai, Searching for non-Abelian phases in the Bose-Einstein condensate of dysprosium, Phys. Rev. A 85, 051606(R) (2012).
  11. X. Cui, B. Lian, T. L. Ho, B. L. Lev, and H. Zhai, Synthetic gauge field with highly magnetic lanthanide atoms, Phys. Rev. A 88, 011601(R) (2013).
  12. J. H. Hannay, The Majorana representation of polarization, and the Berry phase of light, J. Mod. Opt. 45, 1001 (1998).
  13. G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto, Extremal quantum states and their Majorana constellations, Phys. Rev. A 92, 031801(R) (2015).
  14. G. Björk, M. Grassl, P. de la Hoz, G. Leuchs, and L. L. Sánchez-Soto, Stars of the quantum universe: extremal constellations on the Poincaré sphere, Phys. Scr. 90, 108008 (2015).
  15. A. Z. Goldberg, P. de la Hoz, G. Björk, A. B. Klimov, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto, Quantum concepts in optical polarization, Adv. Opt. Photonics 13, 1 (2021).
  16. R. Gutiérrez-Cuevas, S. A. Wadood, A. N. Vamivakas, and M. A. Alonso, Modal Majorana Sphere and Hidden Symmetries of Structured-Gaussian Beams, Phys. Rev. Lett. 125, 123903 (2020).
  17. R. Gutiérrez-Cuevas and M. A. Alonso, Platonic Gaussian beams: wave and ray treatment, Opt. Lett. 45, 6759 (2020).
  18. C. Chryssomalakos and H. Hernández-Coronado, Optimal quantum rotosensors, Phys. Rev. A 95, 052125 (2017).
  19. F. Bouchard, P. de la Hoz, G. Bjork, R. W. Boyd, M. Grassl, Z. Hradil, E. Karimi, A. B. Klimov, G. Leuchs, J. Rehacek, and L. L. Sanchez-Soto, Quantum metrology at the limit with extremal Majorana constellations, Optica 4, 1429 (2017).
  20. A. Z. Goldberg and D. F. V. James, Quantum-limited Euler angle measurements using anticoherent states, Phys. Rev. A 98, 032113 (2018).
  21. A. Z. Goldberg, A. B. Klimov, G. Leuchs, and L. L. Sánchez-Soto, Rotation sensing at the ultimate limit, J. Phys.: Photonics 3, 022008 (2021).
  22. J. H. Hannay, The Berry phase for spin in the Majorana representation, J. Phys. A: Math. Gen. 31, L53 (1998).
  23. P. Bruno, Quantum Geometric Phase in Majorana's Stellar Representation: Mapping onto a Many-Body Aharonov-Bohm Phase, Phys. Rev. Lett. 108, 240402 (2012).
  24. C. Chryssomalakos, E. Guzmán-González, and E. Serrano-Ensástiga, Geometry of spin coherent states, J. Phys. A: Math. Theor. 51, 165202 (2018).
  25. X. Gao, M. Erhard, A. Zeilinger, and M. Krenn, Computer-Inspired Concept for High-Dimensional Multipartite Quantum Gates, Phys. Rev. Lett. 125, 050501 (2020).
  26. H.-H. Lu, Z. Hu, M. S. Alshaykh, A. J. Moore, Y. Wang, P. Imany, A. M. Weiner, and S. Kais, Quantum phase estimation with time-frequency qudits in a single photon, Adv. Quantum Technol. 3, 1900074 (2020).
  27. S. D. Bartlett, H. de Guise, and B. C. Sanders, Quantum encodings in spin systems and harmonic oscillators, Phys. Rev. A 65, 052316 (2002).
  28. M. R. A. Adcock, P. Høyer, and B. C. Sanders, Quantum computation with coherent spin states and the close Hadamard problem, Quantum Inf. Process. 15, 1361 (2016).
  29. A. B. Klimov, R. Guzmán, J. C. Retamal, and C. Saavedra, Qutrit quantum computer with trapped ions, Phys. Rev. A 67, 062313 (2003).
  30. Z. Gedik, I. A. Silva, B. Çakmak, G. Karpat, E. L. G. Vidoto, D. O. Soares-Pinto, E. R. de Azevedo, and F. F. Fanchini, Computational speed-up with a single qudit, Sci. Rep. 5, 14671 (2015).
  31. M. N. Leuenberger and D. Loss, Quantum computing in molecular magnets, Nature (London) 410, 789 (2001).
  32. M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, Twisted photons: New quantum perspectives in high dimensions, Light: Sci. Appl. 7, 17146 (2018).
  33. H. Bechmann-Pasquinucci and A. Peres, Quantum Cryptography with 3-State Systems, Phys. Rev. Lett. 85, 3313 (2000).
  34. D. Bruß and C. Macchiavello, Optimal Eavesdropping in Cryptography with Three-Dimensional Quantum States, Phys. Rev. Lett. 88, 127901 (2002).
  35. Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Front. Phys. 8, 589504 (2020).
  36. D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits, Phys. Rev. Lett. 85, 4418 (2000).
  37. D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Bell Inequalities for Arbitrarily High-Dimensional Systems, Phys. Rev. Lett. 88, 040404 (2002).
  38. T. Vértesi, S. Pironio, and N. Brunner, Closing the Detection Loophole in Bell Experiments Using Qudits, Phys. Rev. Lett. 104, 060401 (2010).
  39. N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures, Phys. Rev. X 5, 041006 (2015).
  40. P. Carruthers and M. M. Nieto, Phase and angle variables in quantum mechanics, Rev. Mod. Phys. 40, 411 (1968).
  41. S. D. Augustin and H. Rabitz, Action-angle variables in quantum mechanics, J. Chem. Phys. 71, 4956 (1979).
  42. R. Lynch, The quantum phase problem: A critical review, Phys. Rep. 256, 367 (1995).
  43. V. Peřinova, A. Lukš, and J. Peřina, Phase in Optics (World Scientific, Singapore, 1998).
  44. A. Luis and L. L. Sánchez-Soto, Quantum phase difference, phase measurements and quantum Stokes parameters, Prog. Opt. 41, 421 (2000).
  45. S. M. Barnett and D. T. Pegg, Quantum theory of rotation angles, Phys. Rev. A 41, 3427 (1990).
  46. A. Luis and L. L. Sánchez-Soto, Quantum theory of rotation angles: The problem of angle sum and angle difference, Eur. Phys. J. D 3, 195 (1998).
  47. J. P. Gazeau and F. H. Szafraniec, Three paths toward the quantum angle operator, Ann. Phys. 375, 16 (2016).
  48. P. Kasperkovitz and M. Peev, Wigner-Weyl formalisms for toroidal geometries, Ann. Phys. 230, 21 (1994).
  49. J. F. Plebański, M. Przanowski, J. Tosiek, and F. J. Turrubiates, Remarks on deformation quantization on the cylinder, Acta Phys. Pol. B 31, 561 (2000).
  50. J. González, M. A. del Olmo, and J. Tosiek, Quantum mechanics on the cylinder, J. Opt. B: Quantum Semiclass. Opt. 5, S306 (2003).
  51. N. Mukunda, Wigner distribution for angle coordinates in quantum mechanics, Am. J. Phys. 47, 182 (1979).
  52. M. Ruzzi and D. Galetti, Schwinger and Pegg–Barnett approaches and a relationship between angular and Cartesian quantum descriptions: II. Phase spaces, J. Phys. A: Math. Gen. 35, 4633 (2002).
  53. N. Mukunda, G. Marmo, A. Zampini, S. Chaturvedi, and R. Simon, Wigner–Weyl isomorphism for quantum mechanics on Lie groups, J. Math. Phys. 46, 012106 (2005).
  54. H. A. Kastrup, Wigner functions for the pair angle and orbital angular momentum, Phys. Rev. A 94, 062113 (2016).
  55. K. Kowalski and K. Ławniczak, Wigner functions and coherent states for the quantum mechanics on a circle, J. Phys. A: Math. Theor. 54, 275302 (2021).
  56. K. Kowalski, J. Rembieliński, and L. C. Papaloucas, Coherent states for a quantum particle on a circle, J. Phys. A: Math. Gen. 29, 4149 (1996).
  57. J. A. González and M. A. del Olmo, Coherent states on the circle, J. Phys. A: Math. Gen. 31, 8841 (1998).
  58. Y. Ohnuki and S. Kitakado, Fundamental algebra for quantum mechanics on SD and gauge potentials, J. Math. Phys. 34, 2827 (1993).
  59. B. C. Hall and J. J. Mitchell, Coherent states on spheres, J. Math. Phys. 43, 1211 (2002).
  60. M. Ruzzi, M. A. Marchiolli, E. C. da Silva, and D. Galetti, Quasiprobability distribution functions for periodic phase spaces: I. Theoretical aspects, J. Phys. A: Math. Gen. 39, 9881 (2006).
  61. H. A. Kastrup, Quantization of the canonically conjugate pair angle and orbital angular momentum, Phys. Rev. A 73, 052104 (2006).
  62. I. Rigas, L. L. Sanchez-Soto, A. B. Klimov, J. Rehacek, and Z. Hradil, Orbital angular momentum in phase space, Ann. Phys. 326, 426 (2011).
  63. H. Bacry, Group Theory and Constellations (Publibook, Paris, 2004).
  64. I. Bengtsson and K. Życzkowski, Geometry of Quantum States (Cambridge University Press, Cambridge, UK, 2017).
  65. A. Z. Goldberg, A. B. Klimov, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto, Extremal quantum states, AVS Quantum Sci. 2, 044701 (2020).
  66. E. C. Lerner, Harmonic-oscillator phase operators, Nuovo Cimento B 56, 183 (1968).
  67. Z. Hradil, J. Řeháček, Z. Bouchal, R. Čelechovský, and L. L. Sánchez-Soto, Minimum Uncertainty Measurements of Angle and Angular Momentum, Phys. Rev. Lett. 97, 243601 (2006).
  68. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer, New York, 1972).
  69. A. Perelomov, Generalized Coherent States and their Applications (Springer, Berlin, 1986).
  70. A. O. Barut and L. Girardello, New “coherent” states associated with non-compact groups, Commun. Math. Phys. 21, 41 (1971).
  71. J.-P. Gazeau and R. Murenzi, Integral quantization for the discrete cylinder, Quantum Rep. 4, 362 (2022).
  72. J.-P. Gazeau, Coherent States in Quantum Physics (Wiley, New York, 2009), p. 360.
  73. P. Dirichlet, Sur la convergence des séries trigonometriques qui servent à représenter une fonction arbitraire entre des limites données, J. Math. 1829, 157 (1829).
  74. L. Fejér, Untersuchungen über Fouriersche Reihen, Math. Ann. 58, 51 (1904).
  75. L. Mišta, H. de Guise, J. Řeháček, and Z. Hradil, Angle and angular momentum: Uncertainty relations, simultaneous measurement, and phase-space representation, Phys. Rev. A 106, 022204 (2022).
  76. K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).
  77. I. Nape, B. Sephton, P. Ornelas, C. Moodley, and A. Forbes, Quantum structured light in high dimensions, APL Photonics 8, 051101 (2023).
  78. W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1986).
  79. U. Chabaud, D. Markham, and F. Grosshans, Stellar Representation of Non-Gaussian Quantum States, Phys. Rev. Lett. 124, 063605 (2020).
  80. R. L. Hudson, When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249 (1974).
  81. I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Non-negative Wigner functions for orbital angular momentum states, Phys. Rev. A 81, 012101 (2010).
  82. D. Mumford, Tata Lectures on Theta I (Birkhauser, Boston, 1983).
  83. DLMF, NIST Digital Library of Mathematical Functions, edited by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, http://dlmf.nist.gov/, Release 1.1.8 of 2022-12-15.
  84. K. Blum, Density Matrix Theory and Applications (Plenum, New York, 1981).
  85. U. Fano and G. Racah, Irreducible Tensorial Sets (Academic, New York, 1959).
  86. A. Z. Goldberg, A. B. Klimov, H. de Guise, G. Leuchs, G. S. Agarwal, and L. L. Sánchez-Soto, From polarization multipoles to higher-order coherences, Opt. Lett. 47, 477 (2022).
  87. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  88. F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Quantum walks and wavepacket dynamics on a lattice with twisted photons, Sci. Adv. 1, e1500087 (2015).
  89. Y. A. Pashkin, O. Astafiev, T. Yamamoto, Y. Nakamura, and J. S. Tsai, Josephson charge qubits: A brief review, Quantum Inf. Process. 8, 55 (2009).
  90. L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, Wigner distribution function for Euclidean systems, J. Phys. A: Math. Gen. 31, 3875 (1998).

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Phys. Rev. Research 5, L032006– Published 12 July, 2023
  • Received 17 March 2023
  • Revised 29 May 2023
  • Accepted 28 June 2023
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