ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Toronov V. Y., Derbov V. L., Priyutova O. M. Geometric phases in the dynamics of nonlinear optical systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 1996, vol. 4, iss. 6, pp. 3-32. DOI: 10.18500/0869-6632-1996-4-6-3-32

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 1996no6p003.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Article
UDC: 
621.375.7:530.182:530.145.6+535.51
DOI: 
10.18500/0869-6632-1996-4-6-3-32

Geometric phases in the dynamics of nonlinear optical systems

Autors: 
Toronov Vladislav Yurevich, Saratov State University
Derbov Vladimir Leonardovich, Saratov State University
Priyutova Olga Mikhailovna, Saratov State University
Abstract: 

Fundamentals of geometric phase theory in general-type dynamical systems and some particular results on geometric phases in semiclassical and quantum models of laser physics and nonlinear optics are reviewed. The emphasis is made оп the physical meaning rather than on formal mathematical background to make the presentation clear for a wide circle of researchers in the field of nonlinear dynamics.

Key words: 
-
Acknowledgments: 
The authors are grateful to V.P. Karasev for cooperation and assistance, as well as A.V. Masalov and A.V. Gorokhov for stimulating discussions and valuable comments. This work was carried out with the financial support of the State Committee for Higher Education of the Russian Federation (grant 95-0-2.1-59).
Reference: 
  1. Berry MV. Quantal phase factors accompanying adiabatic changes. Рrос. Roy. Soc. А. London. 1984;392:45-57. 10.1098/rspa.1984.0023.
  2. Born М, Fock V. Beweis des Adiabatensatzes. Z. Physik. 1928;51:165-180. DOI: 10.1007/BF01343193.
  3. Bitter T, Dubbers D. Manifestation of Berry’s topological phase on neutron spin rotationn. Phys. Rev. Lett. 1987;59(3):251-254. DOI: 10.1103/PhysRevLett.59.251.
  4. Zwanziger JV, Rucker SP, Chingas GC. Measuring the geometric component of the transition probability in а two-level system. Phys. Rev. А. 1991;43(7):3232-3240. DOI: 10.1103/PhysRevA.43.3232.
  5. Suter D, Muller KT, Pines А. Study of the Aharonov-Anandan quantum phase by NMR interferometry. Phys. Rev. Lett. 1988;60(13):1218-1220. DOI: 10.1103/PhysRevLett.60.1218.
  6. Simon В. Holonomy, the quantum adiabatic theorem, and Berry’s Phase. Phys. Rev. Lett. 1983;51(24):2167-2170. DOI: 10.1103/PhysRevLett.51.2167.
  7. Born M, Wolf E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge: Cambridge University Press; 1999. 952 p.
  8. Bhandari R, Samuel J. Observation of Topological Phase by use of а laser interferometer. Phys. Rev. Lett. 1988;60(13):1211-1213. DOI: 10.1103/PhysRevLett.60.1211. Bhandari R. Evolution of light beams in polarization and direction. Physica B: Cond. Matter. 1991;175(1-3):111-122. DOI: 10.1016/0921-4526(91)90699-F.
  9. Markovski BL, Vinitsky SI, editoors. Topological Phases in Quantum Theory. Singapore: World Scientific; 1989.
  10. Shapere А, Wilczek F, editors. Geometric Phases in Physics. Singapore: World Scientific; 1989. 455 p.
  11. Vinitskii SI, Derbov VL, Dubovik VM, Markovski BL, Stepanovskii YuP. Topological phases in quantum mechanics and polarization optics. Sov. Phys. Usp. 1990;33(6):403-428. DOI: 10.1070/PU1990v033n06ABEH002598.
  12. Klyshko DN. Berry geometric phase in oscillatory processes. Phys. Usp. 1993;36(11):1005–1019. DOI: 10.1070/PU1993v036n11ABEH002178.
  13. Ning СZ, Haken Н. The geometric phase in nonlinear dissipative systems. Mod. Phys. Lett. В. 1992;6(25):1541-1568. DOI: 10.1142/s0217984992001265.
  14. Aharonov Y, Anandan J. Phase change during а cyclic quantum evolution. Phys. Rev. Lett. 1987;58(16):1593-1596. DOI: 10.1103/PhysRevLett.58.1593.
  15. Garrison JC, Chiao RY. Geometrical phase from global gauge invariance of nonlinear classical field theories. Phys. Rev. Lett. 1988;60(3):165-168. DOI: 10.1103/PhysRevLett.60.165.
  16. Ning SZ, Haken H. Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors. Phys. Rev. Lett. 1992;68(14):2109-2112. DOI: 10.1103/PhysRevLett.68.2109.
  17. Toronov VYu, Derbov VL. Geometric phases in lasers and liquid flows. Phys. Rev. А. 1994;49(2):1392-1399. DOI: 10.1103/PhysRevA.49.1392.
  18. Chaika МP. Interference of Degenerate Atomic States. Leningrad: Leningrad University Publishing; 1975. 192 p.
  19. Dubrovin BA, Novikov SP, Fomenko LТ. Modern Geometry: Methods of Homology Theory. M.: Nauka; 1984. 343 p.
  20. Pancharatnam S. Generalized theory of interference, and its applications. Рrос. Ind. Acad. Sci. А. 1956;44:247-262. DOI: 10.1007/BF03046050.
  21. Samuel J, Bhandari R. General setting for Berry’s phase. Phys. Rev. Lett. 1988;60(23):2339-2342. DOI: 10.1103/PhysRevLett.60.2339.
  22. Toronov VYu, Derbov VL. Geometric phases in a ring laser. Quantum Electron. 1997;27(7):644-648. DOI: 10.1070/QE1997v027n07ABEH001007.
  23. Sargent M. III, Scully МО, Lamb WE.  Laser Physics. London: Addison-Wesley, 1974. 464 p.
  24. Haken H. Laser Theory. Berlin: Springer; 1984. 322 p. DOI: 10.1007/978-3-642-45556-8.
  25. Weiss СО, Abraham NB, Hubner U. Homoclinic and heteroclinic chaos in а single-mode laser. Phys. Rev. Lett. 1988;61(14):1587-1590. DOI: 10.1103/PhysRevLett.61.1587.
  26. Ning CZ, Haken H. Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations. Phys. Rev. А. 1990;41(7):3826-3837. DOI: 10.1103/physreva.41.3826.
  27. Vladimirov АG, Toronov VYu, Derbov VL. About the complex Lorentz model. Izvestiya VUZ. Applied Nonlinear Dynamics . 1995;3(6):51.
  28. Roldan E, dе Valcarcel GJ, Vilaseca R, Mandel P. Single-mode-laser phase dynamics. Phys. Rev. A. 1993;48(1):591-598. DOI: 10.1103/physreva.48.591.
  29. Fowler AC, Gibbon JD, McGuinness MJ. The complex Lorenz equations. Physica D. 1982;4(2):139-163. DOI: 10.1016/0167-2789(82)90057-4. Gibbon JD, McGuinness MJ. The real and complex Lorenz equations in rotating fluids and lasers. Physica D. 1982;5(1):108-122. DOI: 10.1016/0167-2789(82)90053-7.
  30. Ning CZ, Haken H. Quaziperiodicity involving twin oscillations in the complex Lorenz equations describing а detuned laser. Z. Physik В. 1990;81:457-461. DOI: 10.1007/BF01390829.
  31. Vilaseca R, de Valcarcel GJ, Roldan E. Physical interprhtion of laser phase dynamics. Phys. Rev. А. 1990;41(9):5269-5272. DOI: 10.1103/PhysRevA.41.5269.
  32. Toronov VYu, Derbov VL. Geometric phase effects in laser dynamics. Phys. Rev. А. 1994;50(1):878-881. DOI: 10.1103/PhysRevA.50.878.
  33. Chyba TH. Phase-jump instability in the bidirectional ring laser with backscattering. Phys. Rev. А. 1989;40(11):6327-6338. DOI: 10.1103/physreva.40.6327.
  34. Skryabin DV, Vladimirov AD, Radin AM. Spontaneous phase symmetry breaking due to cavity detuning in а class-A bidirectional ring laser. Орt. Comm. 1995;116(1-3):109-115. DOI: 10.1016/0030-4018(95)00050-i.
  35. Etrich C, Mandel Р, Neelen RC, Spreeuw RJC, Woerdman JP. Dynamics of a ring-laser gyroscope with backscattering. Phys. Rev. А. 1992;46(1):525-536. DOI: 10.1103/physreva.46.525.
  36. Hoffer LM, Abraham NB. Analysis of а coherent model for а homogeneously broadened bidirectional ring laser. Opt. Comm. 1990;74(3-4):261-268. DOI: 10.1016/0030-4018(89)90360-X.
  37. Abraham NB, Weiss СО. Dynamical frequency shifts and intensity pulsations in an FIR bidirectional ring laser. Opt. Comm. 1988;68(6):437-441. DOI: 10.1016/0030-4018(88)90248-9.
  38. Khandokhin РА, Khanin Yal. Instabilities in a solid-state ring laser. JOSA B. 1985;2(1):226-231. DOI: 10.1364/JOSAB.2.000226.
  39. Koryukin NV, Khandokhin PA, Khanin YaI. Frequency dynamics of a bidirectional ring laser with a nonreciprocal resonator. Sov. J. Quantum Electron. 1990;20(8):895-898. DOI: 10.1070/QE1990v020n08ABEH007152.
  40. Klimontovich YuL, Kuryatov VN, Landa PS. Wave Synchronization in a Gas Laser with a Ring Resonator. J. Exp. Theor. Phys. 1966;24(1):1-7.
  41. Messiah А. Quantum Mechanics. Mineola: Dover Publications; 1999. 1136 p.
  42. Perelomov АМ. Generalised Coherent States and Their Applications. M.: Nauka; 1987. 268 p.
  43. Teich MC, Saleh BEA. Squeezed states of light. Quantum Opt. 1989;1:153-191.
  44. Akhmanov SA, Belinskii AV, Chirkin АS. Phase bistability and multistability in concentrated and distributed systems: classical and quantum aspects. In: Akhmanov SA, Vorontsov MA, editors. New Physical Principles of Optical Information Processing. M.: Nauka; 1990. P. 83.
  45. Smirnov DF, Troshin АS. New phenomena in quantum optics: photon antibunching, sub-Poisson photon statistics, and squeezed states. Sov. Phys. Usp. 1987;30(10):851-874. DOI: 10.1070/PU1987v030n10ABEH002966.
  46. Bykov VP. Squeezed light and nonclassical motion in mechanics. Phys. Usp. 1993;36(9):841-850. DOI: 10.1070/pu1993v036n09abeh002309.
  47. Klyshko DN, Masalov АV. Photon noise: observation, squeezing, interpretation. Phys. Usp. 1995;38(11):1203-1230. DOI: 10.1070/pu1995v038n11abeh000117.
  48. Kitano М, Yabuzaki Т. Observation of Lorentz-group Berry phases in polarization optics. Phys. Lett. 1989;142(6-7):321-325. DOI: 10.1016/0375-9601(89)90373-3.
  49. Golubev YuМ. Statistics of an electromagnetic field parametrically interacting with the medium. Opt. Spectroscopy. 1979;46(2):398-400.
  50. Wodkiewicz K, Zubairy MS. Effect of laser fluctuations on squeezed states in a degenerate parametric amplifier. Phys. Rev. А. 1983;27(4):2003-2007. DOI: 10.1103/PhysRevA.27.2003.
  51. Gorbachev VN, Zanadvorov PN. Quantum statistics of the second harmonic generation process. Opt. Spectroscopy. 1980;49(3):600-605.
  52. Milburn С, Walls DF. Production of squeezed states in а degenerate раrametric amplifier. Opt. Comm. 1981;39(6):401-404. DOI: 10.1016/0030-4018(81)90232-7.
  53. Becker W, Scully МО, Zubuiry MS. Generation of а Squeezed Coherent States via а Free-Electron Laser. Phys. Rev. Lett. 1982;48(7):475-477. DOI: 10.1103/PhysRevLett.48.475.
  54. Sibilia G, Bertolorti M, Peurinov'a V, Peurina J, Lukus А. Photon antibunching effect and statistical properties of single-mode emission in free-electron lasers. Phys. Rev. А. 1983;28(1):328-331. DOI: 10.1103/PhysRevA.28.328.
  55. Rai Suranjana, Chopra S. Photon statistics and squeezing properties of a free-electron laser. Phys. Rev. А. 1984;30(4):2104-2107. DOI: 10.1103/PhysRevA.30.2104.
  56. Tanzler W, Schutte FJ. Antibuncing in multiphoton processes. Opt. Comm. 1981;37(6):447-450. DOI: 10.1016/0030-4018(81)90139-5.
  57. Slusher RE, Yurke B, Valley JF. Experimental study of squeezed states using four-wave mixing in а cavity configuration. JOSA B. 1984;1(3):525.
  58. Reid MD, Walls DF. Generation of squeezed states via degenerate four-wave mixing. Phys. Rev. А. 1985;31(3):1622-1635. DOI: 10.1103/PhysRevA.31.1622.
  59. Slusher RE, Yurke B, Grandgier Р, Lа Porta А, Walls DF, Reid M. Squeezed-light generation by four-wave mixing near an atomic resonance. JOSA В. 1987;4(10):1453-1464. DOI: 10.1364/JOSAB.4.001453.
  60. Milburg GJ, Walls DF, Levenson MD. Quantum phase fluctuations and squeezing in degenerate four-wave mixing. JOSA В. 1984;1(3):390-394. DOI: 10.1364/JOSAB.1.000390.
  61. Levenson MD, Shelby RM, Aspect А, Reid M, Walls DF. Generation and detection of squeezed states of light by nongenerate four-wave mixing in an optic fiber. Phys. Rev. А. 1985;32(3):1550-1562. DOI: 10.1103/physreva.32.1550.
  62. Slusher RE, Hollberg LW, Yurke B, Mertz JC, Valley JF. Observation of squeezed states generated by four-wave-mixing in an optical cavity. Phys. Rev. Lett. 1985;55(22):2409-2412. DOI: 10.1103/PhysRevLett.55.2409.
  63. Korolkova NV, Chirkin АS. Source of intense radiation with nonclassical properties. Quantum Electron. 1994;24(12):1027-1028. DOI: 10.1070 QE1994v024n12ABEH000255 Korolkova NV, Chirkin АS. Doubling the frequency of coherent radiation in media with quadratic and cubic optical nonlinearities; approximation of a given number of photons. Opt. Spectroscopy. 1993;74(5.):908.
  64. Loisell WH. Radiation and Noise in Quantum Electronics. N.Y.: McGraw-Hill; 1964. 303 p.
  65. Damaskinskii EV. Calculation of the Berry phase in compressed states. Notes of Scientific Seminars of the Leningrad Branch of the Mathematical Institute. Questions of quantum field theory and statistical physics. 1989;169(8):51-59.
  66. Karassiov VP, Derbov VL, Vinitsky SI. Polarization coherent states in quantum optics: quasiprobability functions and geometric phases. Proc. SPIE. 1993;2098:164. DOI: 10.1117/12.175142.
  67. Karassiov VP, Derbov VL, Privutova OM, Vinitsky SI. Geometric Phases of the Generelized Polarization Coherent States in Quantum Optics. In: VII International Conference on Symmetry methods in physics. Abstracts. 1995, Dubna, Russia. P. 27.
  68. Karassiov VP. Polarization structure of quantum light: а new insight. 2: Generalized coherent states, squeezing and geometric phases. Е-ргт Аrсvе QUANT PH/9503011, 1995.
  69. Karassiov VP. Polarization structure of quantum light: a new insight. 1: General outlook. J. Phys. А: Math. Gen. 1993;26:4345-4354. DOI: 10.1088/0305-4470/26/17/040.
  70. Karassiov VP, Masalov AV. States of non-polarised light in quantum optics. Opt. Spectroscopy. 1993;74(5):928-936.
  71. Karassiov VP. Polarization squeezing and new states of light in quantum optics. Phys. Lett. А. 1994;190(5-6):387-392. DOI: 10.1016/0375-9601(94)90720-X.
  72. Berry M. The adiabatic phase and Pancharatnam's phase for polarized light. J. Mod. Opt. 1987;34(11):1401-1407.
  73. Chyba TH, Wang LJ, Mandel L, Simon R. Measurement of the Pancharatnam phase for a light beam. Opt. Lett. 1988;13:562. DOI: 10.1364/ol.13.000562.
  74. Jordan TF Веrrу phases and unitary transformations. J. Math. Phys. 1988;29(9):2042-2052. DOI: 10.1063/1.527862.
  75. Klyshko DN. Веrrу phase in multyphoton experiments. Phys. Lett. А. 1989;140(1-2):19-24. DOI: 10.1016/0375-9601(89)90539-2.
  76. Chiao RY, Jordan TF. Lorentz-group Berry phases in squeezed light. Phys. Lett. А. 1988;132(2-3):77-81. DOI: 10.1016/0375-9601(88)90255-1.
  77. Agarwal GS, Simon R. Berry phase, interference of light beams, and the Hannay angle. Phys. Rev. А. 1990;42(11):6924-6927. DOI: 10.1103/PhysRevA.42.6924.
  78. Hannay JH. Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A: Math. Gen. 1985;18(2):221-230. DOI: 10.1088/0305-4470/18/2/011. Веrrу MV. Classical adiabatic angles and quantal adiabatic phase. J. Phys. A: Math. Gen. 1985;18(1):15-27. DOI: 10.1088/0305-4470/18/1/012.
  79. Maamache M, Provost J-P, Vallee G. Berry's phase and Hannay's angle from quantum canonical transformations. J. Phys. A: Math. Gen. 1991;24(3):685-688. DOI: 10.1088/0305-4470/24/3/027.
  80. Giavarini G, Gozzi E, Rohrlich D, Thacker WD. Some connections between classical and quantum anholonomy. Phys. Rev. D. 1989;39(10):3007-3015. DOI: 10.1103/PhysRevD.39.3007.
  81. Zhelobenko DP, Shtern АI. Representations of Lee Groups. M.: Nauka; 1983. 360 p.
  82. Ott E, Grebogi C, Yorke J.A. Controlling Chaos. Phys. Rev. Lett. 1990;64(11):1196-1199. DOI: 10.1103/PhysRevLett.64.1196.
  83. Arnold VI. Mathematical Methods of Classical Mechanics. M.: Nauka; 1974; 431 p. Arnold VI. Catastrophe theory. М.: Nauka; 1990. 126 p. Abraham R, Marsden JE. Foundations of Mechanics. Reading: Benjamin; 1978. 806 p.
Received: 
14.06.1996
Accepted: 
02.12.1996
Published: 
08.04.1997
  • 257 reads