Frontiers of Physics

, Volume 7, Issue 3, pp 261–310 | Cite as

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

Review Article
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Abstract

By virtue of the new technique of performing integration over Dirac’s ket-bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.

Keywords

Dirac’s symbolic method IWOP technique entangled state of continuum variables entangled Fresnel transform Collins formula Generalized Fresnel operator complex wavelet transform complex Wigner transform complex fractional Fourier transform symplectic wavelet transform entangled symplectic wavelet transform Symplectic-dilation mixed wavelet transform fractional Radon transform new eigenmodes of fractional Fourier transform 
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References

  1. 1.
    R. J. Glauber, Phys. Rev., 1963, 130: 2529MathSciNetADSGoogle Scholar
  2. 2.
    R. J. Glauber, Phys. Rev., 1963, 131: 2766MathSciNetADSGoogle Scholar
  3. 3.
    J. R. Klauder, Ann. Phys., 1960, 11: 123MathSciNetADSMATHGoogle Scholar
  4. 4.
    J. R. Klauder and B.-S. Skagerstam, Coherent States, Singapore: World Scientific, 1985MATHGoogle Scholar
  5. 5.
    J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics, New York: W. A. Benjamin Inc., 1968Google Scholar
  6. 6.
    E. Schrödinger, Naturwiss, 1926, 14: 664ADSGoogle Scholar
  7. 7.
    P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd, Oxford: Clarendon Press, 1930MATHGoogle Scholar
  8. 8.
    R. J. Glauber, in: Optique et Electronique Quantiques — Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji (Eds.), New York: Gordon and Breach, 1965Google Scholar
  9. 9.
    L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge: Cambridge University Press, 1996Google Scholar
  10. 10.
    M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge: Cambridge University Press, 1997Google Scholar
  11. 11.
    D. F. Walls and G. J. Milburn, Quantum Optics, Berlin: Springer-Verlag, 1994MATHGoogle Scholar
  12. 12.
    G. Nienhuis and L. Allen, Phys. Rev. A, 1993, 48: 656ADSGoogle Scholar
  13. 13.
    D. Dragoman, Progress in Optics, 2002, 42: 424, and references thereinGoogle Scholar
  14. 14.
    H. P. Yuen, Phys. Rev. A, 1976, 13: 2226ADSGoogle Scholar
  15. 15.
    R. Loudon and P. L. Knight, J. Mod. Opt., 1987, 34: 709MathSciNetADSMATHGoogle Scholar
  16. 16.
    D. F. Walls, Nature, 1983, 306: 141ADSGoogle Scholar
  17. 17.
    M. C. Teich and B. E. A. Saleh, Quantum Opt., 1989, 1: 153ADSGoogle Scholar
  18. 18.
    H. Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D, 1987, 35: 1831MathSciNetADSGoogle Scholar
  19. 19.
    H. Y. Fan and H. R. Zaidi, Phys. Rev. A, 1988, 37: 2985ADSGoogle Scholar
  20. 20.
    H. Y. Fan and J. R. Klauder, J. Phys. A, 1988, 21: L715MathSciNetGoogle Scholar
  21. 21.
    H. Y. Fan and H. Zou, Phys. Lett. A, 1999, 252: 281MathSciNetADSMATHGoogle Scholar
  22. 22.
    H. Y. Fan, J. Opt. B, 2003, 5: R147ADSGoogle Scholar
  23. 23.
    H. Y. Fan, Int. J. Mod. Phys. B, 2004, 18: 1387ADSMATHGoogle Scholar
  24. 24.
    H. Y. Fan, Int. J. Mod. Phys. B, 2004, 18: 2771ADSMATHGoogle Scholar
  25. 25.
    H. Y. Fan and J. R. Klauder, Phys. Rev. A, 1994, 49: 704ADSMATHGoogle Scholar
  26. 26.
    H. Y. Fan and X. Ye, Phys. Rev. A, 1995, 51: 3343ADSGoogle Scholar
  27. 27.
    H. Y. Fan and A. Wünsche, J. Opt. B, 2000, 2: 464ADSGoogle Scholar
  28. 28.
    H. Y. Fan, H. L. Lu, and Y. Fan, Ann. Phys., 2006, 321: 480MathSciNetADSMATHGoogle Scholar
  29. 29.
    A. Wünsche, J. Opt. B, 1999, 1: R11Google Scholar
  30. 30.
    M. Born and E. Wolf, Principles of Optics, 7th edition, Singapore: World Scientific, 1999Google Scholar
  31. 31.
    J. W. Goodman, Introduction to Fourier Optics, New York: McGraw-Hill, 1972Google Scholar
  32. 32.
    A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev., 1935, 47: 777ADSMATHGoogle Scholar
  33. 33.
    H. Y. Fan, Phys. Lett. A, 2003, 313: 343MathSciNetADSMATHGoogle Scholar
  34. 34.
    S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics, 3rd Ed., Cambridge: Cambridge University Press, 1998Google Scholar
  35. 35.
    A. Torre, FrFT and Its Applications in Optics, Progress in Optics, Vol. 43, edited by E. Wolf, 2003, and references thereinGoogle Scholar
  36. 36.
    V. Namias, J. Inst. Math. Appl., 1980, 25: 241MathSciNetMATHGoogle Scholar
  37. 37.
    D. Mendlovic and H. M. Ozakatas, J. Opt. Soc. Am. A, 1993, 10: 1875ADSGoogle Scholar
  38. 38.
    D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt., 1994, 33: 6188ADSGoogle Scholar
  39. 39.
    H. M. Ozakatas and D. Mendlovic, J. Opt. Soc. Am. A, 1993, 10: 2522ADSGoogle Scholar
  40. 40.
    A. W. Lohmann, J. Opt. Soc. Am. A, 1993, 10: 2181ADSGoogle Scholar
  41. 41.
    L. M. Bernardo and O. D. Soares, Opt. Commun., 1994, 110: 517ADSGoogle Scholar
  42. 42.
    E. Wigner, Phys. Rev., 1932, 40: 749ADSMATHGoogle Scholar
  43. 43.
    S. A. Collins, J. Opt. Soc. Am., 1970, 60: 1168ADSGoogle Scholar
  44. 44.
    S. Wang and D. M. Zhao, Metrix Optics, Berlin Heidelberg: Springer-Verlag, and Beijing: Higher Education Press, 2000Google Scholar
  45. 45.
    J. A. Arnaud, J. Opt. Soc. Am., 1971, 61: 751ADSGoogle Scholar
  46. 46.
    C. Gomez-Reino, Int. J. Optoelectron., 1992, 7: 607Google Scholar
  47. 47.
    O. Seger, Ph. D. Dissertation 301, Linkoping University, 1993Google Scholar
  48. 48.
    T. Alieva and F. Agullo-Lopez, Opt. Commun., 1995, 114: 161ADSGoogle Scholar
  49. 49.
    T. Alieva and F. Agullo-Lopez, Opt. Commun., 1996, 115: 267Google Scholar
  50. 50.
    D. F. V. James and G. S. Agarwal, Opt. Commun., 1996, 126: 207ADSGoogle Scholar
  51. 51.
    H. Weyl, Z. Phys., 1927, 46: 1ADSGoogle Scholar
  52. 52.
    H. Y. Fan and T. Q. Song, J. Phys. A, 2003, 36: 7803MathSciNetADSMATHGoogle Scholar
  53. 53.
    H. Y. Fan, Phys. Rev. A, 2002, 65: 064102MathSciNetADSGoogle Scholar
  54. 54.
    H. Y. Fan, Phys. Lett. A, 2002, 294: 253MathSciNetADSMATHGoogle Scholar
  55. 55.
    H. Y. Fan and Y. Fan, Phys. Rev. A, 1996, 54: 958Google Scholar
  56. 56.
    R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A, 1989, 50: 5274Google Scholar
  57. 57.
    C. Silberhorn, P. K. Lam, O. Weiss, F. König, N. Korolkova, and G. Leuchs, Phys. Rev. Lett., 2001, 86: 4267ADSGoogle Scholar
  58. 58.
    E. C. G. Sudarshan, Phys. Rev. Lett., 1963, 10: 277MathSciNetADSMATHGoogle Scholar
  59. 59.
    H. Y. Fan and B. Z. Chen, Phys. Rev. A, 1996, 53: 1948Google Scholar
  60. 60.
    H. J. Wu and H. Y. Fan, Mod. Phys. Lett. B, 1997, 11: 544MathSciNetADSGoogle Scholar
  61. 61.
    H. Y. Fan, H. Zou, and Y. Fan, Phys. Lett. A, 1999, 254: 137ADSGoogle Scholar
  62. 62.
    D. Casasent and D. Psaltis, Proc. IEEE, 1977, 65: 77Google Scholar
  63. 63.
    H. Y. Fan, Opt. Commun., 2008, 281: 2023ADSGoogle Scholar
  64. 64.
    A. Erdelyi, Higher Transcendental Functions, The Bateman Manuscript Project, New York: McGraw-Hill, 1953Google Scholar
  65. 65.
    H. Y. Fan and H. L. Lu, Phys. Lett. A, 2005, 334: 132ADSMATHGoogle Scholar
  66. 66.
    A. Gerrard and N. M. Burch, Introduction to Matrix Methods in Optics, London: John Wiley & Sons, 1975Google Scholar
  67. 67.
    H. Y. Fan, Commun. Theor. Phys., 2002, 38: 147Google Scholar
  68. 68.
    H. Y. Fan and J. H. Chen, Commun. Theor. Phys., 2003, 40: 589MathSciNetMATHGoogle Scholar
  69. 69.
    H. Y. Fan and Z.-H. Xu, Phys. Rev. A, 1994, 50: 2921MathSciNetGoogle Scholar
  70. 70.
    H. Y. Fan and J. VanderLinde, Phys. Rev. A, 1989, 39: 2987MathSciNetADSGoogle Scholar
  71. 71.
    H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics, Shanghai: Shanghai Scientific and Technical Publishers, 1997 (in Chinese)Google Scholar
  72. 72.
    H. Y. Fan, Entangled State Representations in Quantum Mechanics and Their Applications, Shanghai: Shanghai Jiao Tong University Press, 2001Google Scholar
  73. 73.
    H. Kogelnik and T. Li, Appl. Opt., 1966, 5: 1550ADSGoogle Scholar
  74. 74.
    P. A. Bélanger, Opt. Lett., 1991, 16: 196ADSGoogle Scholar
  75. 75.
    V. Magni, G. Cerullo, and S. D. Silvestri, Opt. Commun., 1993, 96: 348ADSGoogle Scholar
  76. 76.
    H. Y. Fan and L. Y. Hu, Opt. Commun., 2008, 281: 1629ADSGoogle Scholar
  77. 77.
    M. Nazarathy and J. Shamir, J. Opt. Soc. Am., 1982, 72: 398Google Scholar
  78. 78.
    M. Nazarathy and J. Shamir, J. Opt. Soc. Am., 1982, 72: 356MathSciNetADSGoogle Scholar
  79. 79.
    M. Nazarathy, A. Hardy, and J. Shamir, J. Opt. Soc. Am. A, 1986, 3: 360MathSciNetGoogle Scholar
  80. 80.
    H. Y. Fan and A. Wünsche, Commun. Theor. Phys., 2003, 39: 717Google Scholar
  81. 81.
    K. Vogel and H. Risken, Phys. Rev. A, 1989, 40: 2847ADSGoogle Scholar
  82. 82.
    W. Vogel and W. P. Schleich, Phys. Rev. A, 1991, 44: 7642ADSGoogle Scholar
  83. 83.
    U. Leonhardt, Measuring the Quantum State of Light, Cambridge: Cambridge University Press, 1997, and references thereinGoogle Scholar
  84. 84.
    M. Aspelmeyer, Nat. Phys., 2009, 5: 11Google Scholar
  85. 85.
    I. L. Chuang and M. A. Nielson, Mod. Opt., 1997, 44: 2455ADSGoogle Scholar
  86. 86.
    W. P. Schleich, Quantumm Optics in Phase Space, Berlin: Wiley-VCH, 2001Google Scholar
  87. 87.
    A. Wünsche, J. Mod. Opt., 1997, 44: 2293ADSMATHGoogle Scholar
  88. 88.
    A. Wünsche, Phys. Rev. A, 1996, 54: 5291ADSGoogle Scholar
  89. 89.
    H. Y. Fan and L. Y. Hu, Opt. Commun., 2009, 282: 3734ADSGoogle Scholar
  90. 90.
    R. F. O’Connell and E. P. Wigner, Phys. Lett. A, 1981, 83: 145MathSciNetADSGoogle Scholar
  91. 91.
    G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2161ADSGoogle Scholar
  92. 92.
    G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2187ADSGoogle Scholar
  93. 93.
    G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2206ADSGoogle Scholar
  94. 94.
    M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep., 1984, 106: 121MathSciNetADSGoogle Scholar
  95. 95.
    A. C. McBride, and F. H. Kerr, IMA J. Appl. Math., 1987, 39: 159MathSciNetMATHGoogle Scholar
  96. 96.
    Y. G. Cai and S. Y. Zhu, Opt. Lett., 2005, 30: 388ADSGoogle Scholar
  97. 97.
    S. Liu, J. Xu, Y. Zhang, L. Chen, and C. Li, Opt. Lett., 1995, 20: 1053ADSGoogle Scholar
  98. 98.
    C. C. Shih, Opt. Lett., 1995, 20: 1178ADSGoogle Scholar
  99. 99.
    L. M. Bernardo and O. D. Soares, Appl. Opt., 1996, 35: 3163ADSGoogle Scholar
  100. 100.
    L. M. Bernardo and O. D. Soares, J. Opt. Soc. Am. A, 1994, 11: 2622ADSGoogle Scholar
  101. 101.
    S. Chountasis, A. Vourdas, and C. Bendjaballah, Phys. Rev. A, 1999, 60: 3467MathSciNetADSGoogle Scholar
  102. 102.
    H. Y. Fan and L. Y. Hu, Chin. Phys. Lett., 2008, 25: 513ADSGoogle Scholar
  103. 103.
    H. M. Ozaktas and M. F. Erden, Opt. Commun., 1997, 143: 75ADSGoogle Scholar
  104. 104.
    H. Y. Fan and L. Y. Hu, J. Mod. Opt., 2009, 56: 1819ADSMATHGoogle Scholar
  105. 105.
    H. Lee, Phys. Rep., 1995, 259: 147MathSciNetGoogle Scholar
  106. 106.
    H. Y. Fan, Opt. Lett., 2003, 28: 2177ADSGoogle Scholar
  107. 107.
    H. Y. Fan and H. L. Lu, Opt. Lett., 2006, 31: 2622ADSGoogle Scholar
  108. 108.
    L. Y. Hu and H. Y. Fan, J. Mod. Opt., 2008, 55: 2429ADSMATHGoogle Scholar
  109. 109.
    H. Y. Fan, L. Y. Hu, and J. S. Wang, J. Opt. Soc. Am. A, 2009, 25: 974ADSGoogle Scholar
  110. 110.
    H. Y. Fan and C. H. Lv, J. Opt. Soc. Am. A, 2009, 26: 2306MathSciNetADSGoogle Scholar
  111. 111.
    D. Dragoman, J. Opt. Soc. Am. A, 2009, 26: 274MathSciNetGoogle Scholar
  112. 112.
    H. Y. Fan, Commun. Theor. Phys., 1999, 31: 285Google Scholar
  113. 113.
    H. Y. Fan and Y. Fan, Commun. Theor. Phys., 2000, 33: 701Google Scholar
  114. 114.
    H. Y. Fan and L. S. Li, Commun. Theor. Phys., 1998, 29: 477Google Scholar
  115. 115.
    P. Pellat-Finet, Opt. Lett., 1994, 19: 1388ADSGoogle Scholar
  116. 116.
    See: e.g., S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology, SIAM, Philadelphia, 2001MATHGoogle Scholar
  117. 117.
    M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Book/Cole, 2002Google Scholar
  118. 118.
    C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction toWavelets and Wavelet Transforms: A Primer, New Jersey: Prentice Hall, 1998Google Scholar
  119. 119.
    C. K. Chiu, Introduction to Wavelets, San Diego: Academic Press, 1992Google Scholar
  120. 120.
    D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett., 1978, 12: 33Google Scholar
  121. 121.
    E. Kapon, J. Matz, and A. Yariv, Opt. Lett., 1984, 10: 125ADSGoogle Scholar
  122. 122.
    M. Oka, H. Masuda, Y. Kaneda, and W. Streifer, IEEE J. Quantum Electron., 1992, 28: 1142ADSGoogle Scholar
  123. 123.
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 1992Google Scholar
  124. 124.
    L. Y. Hu and H. Y. Fan, J. Mod. Opt., 2008, 55: 1835ADSMATHGoogle Scholar
  125. 125.
    P. S. Addison, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance, London: Institute of Physics Publishing, 2002MATHGoogle Scholar
  126. 126.
    P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A, 1995, 51: R1750ADSGoogle Scholar
  127. 127.
    S. De Luca and E. Fiordilino, J. Phys. B, 1996, 29: 3277ADSGoogle Scholar
  128. 128.
    K. Husimi, Proc. Phys.-Math. Soc. Jpn., 1940, 22: 264MATHGoogle Scholar
  129. 129.
    C. L. Mehta, Phys. Rev. Lett., 1967, 18: 752ADSGoogle Scholar
  130. 130.
    H. Y. Fan, Ann. Phys., 2007, doi: 10.1016/j.aop.2007.06.003Google Scholar
  131. 131.
    L. Y. Hu and H. Y. Fan, Int. J. Theor. Phys., 2009, 48: 1539MathSciNetMATHGoogle Scholar
  132. 132.
    H. Y. Fan and H. L. Lu, Opt. Lett., 2006, 31: 3432ADSGoogle Scholar
  133. 133.
    H. Y. Fan and S. G. Liu, Opt. Lett., 2007, 32: 1507ADSGoogle Scholar
  134. 134.
    H. Y. Fan, S. G. Liu, and L. Y. Hu, Opt. Lett., 2009, 34: 551ADSGoogle Scholar
  135. 135.
    H. Y. Fan and H. L. Lu, J. Phys. A, 2004, 37: 10993MathSciNetADSMATHGoogle Scholar
  136. 136.
    H. Y. Fan, X. B. Tan, and H. L. Lu, Phys. Lett. A, 2006, 357: 163ADSGoogle Scholar
  137. 137.
    M. A. Nielsen and I. L. Chuang, The Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000Google Scholar
  138. 138.
    See: e.g., J. Preskill, Quantum Information and Computation, California Institute of Technology, 1998Google Scholar
  139. 139.
    S. Parker, S. Bose, and M. Plenio, Phys. Rev. A, 2000, 61: 032305ADSGoogle Scholar
  140. 140.
    C. M. Xie and H. Y. Fan, J. Mod. Opt., 2010, 57: 582MathSciNetADSMATHGoogle Scholar

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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of PhysicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.College of Physics & Communication ElectronicsJiangxi Normal UniversityNanchangChina
  3. 3.Department of Material Science and EngineeringUniversity of Science and Technology of ChinaHefeiChina

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