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

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

References

[1] R. J. Glauber, Phys. Rev., 1963, 130: 2529

[2] R. J. Glauber, Phys. Rev., 1963, 131: 2766

[3] J. R. Klauder, Ann. Phys., 1960, 11: 123

[4] J. R. Klauder and B.-S. Skagerstam, Coherent States, Singapore: World Scientific, 1985

[5] J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics, New York: W. A. Benjamin Inc., 1968

[6] E. Schrödinger, Naturwiss, 1926, 14: 664

[7] P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd, Oxford: Clarendon Press, 1930

[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, 1965

[9] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge: Cambridge University Press, 1996

[10] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge: Cambridge University Press, 1997

[11] D. F. Walls and G. J. Milburn, Quantum Optics, Berlin: Springer-Verlag, 1994

[12] G. Nienhuis and L. Allen, Phys. Rev. A, 1993, 48: 656

[13] D. Dragoman, Progress in Optics, 2002, 42: 424, and references therein

[14] H. P. Yuen, Phys. Rev. A, 1976, 13: 2226

[15] R. Loudon and P. L. Knight, J. Mod. Opt., 1987, 34-709

[16] D. F. Walls, Nature, 1983, 306: 141

[17] M. C. Teich and B. E. A. Saleh, Quantum Opt., 1989, 1: 153

[18] H. Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D, 1987, 35: 1831

[19] H. Y. Fan and H. R. Zaidi, Phys. Rev. A, 1988, 37: 2985

[20] H. Y. Fan and J. R. Klauder, J. Phys. A, 1988, 21: L715

[21] H. Y. Fan and H. Zou, Phys. Lett. A, 1999, 252: 281

[22] H. Y. Fan, J. Opt. B, 2003, 5: R147

[23] H. Y. Fan, Int. J. Mod. Phys. B, 2004, 18: 1387

[24] H. Y. Fan, Int. J. Mod. Phys. B, 2004, 18: 2771

[25] H. Y. Fan and J. R. Klauder, Phys. Rev. A, 1994, 49: 704

[26] H. Y. Fan and X. Ye, Phys. Rev. A, 1995, 51: 3343

[27] H. Y. Fan and A. Wünsche, J. Opt. B, 2000, 2: 464

[28] H. Y. Fan, H. L. Lu, and Y. Fan, Ann. Phys., 2006, 321: 480

[29] A. Wünsche, J. Opt. B, 1999, 1: R11

[30] M. Born and E. Wolf, Principles of Optics, 7th edition, Singapore: World Scientific, 1999

[31] J. W. Goodman, Introduction to Fourier Optics, New York: McGraw-Hill, 1972

[32] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev., 1935, 47: 777

[33] H. Y. Fan, Phys. Lett. A, 2003, 313: 343

[34] S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics, 3rd Ed., Cambridge: Cambridge University Press, 1998

[35] A. Torre, FrFT and Its Applications in Optics, Progress in Optics, Vol. 43, edited by E. Wolf, 2003, and references therein

[36] V. Namias, J. Inst. Math. Appl., 1980, 25: 241

[37] D. Mendlovic and H. M. Ozakatas, J. Opt. Soc. Am. A, 1993, 10: 1875

[38] D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt., 1994, 33: 6188

[39] H. M. Ozakatas and D. Mendlovic, J. Opt. Soc. Am. A, 1993, 10: 2522

[40] A. W. Lohmann, J. Opt. Soc. Am. A, 1993, 10: 2181

[41] L. M. Bernardo and O. D. Soares, Opt. Commun., 1994, 110: 517

[42] E. Wigner, Phys. Rev., 1932, 40: 749

[43] S. A. Collins, J. Opt. Soc. Am., 1970, 60: 1168

[44] S. Wang and D. M. Zhao, Metrix Optics, Berlin Heidelberg: Springer-Verlag, and Beijing: Higher Education Press, 2000

[45] J. A. Arnaud, J. Opt. Soc. Am., 1971, 61: 751

[46] C. Gomez-Reino, Int. J. Optoelectron., 1992, 7: 607

[47] O. Seger, Ph. D. Dissertation 301, Linkoping University, 1993

[48] T. Alieva and F. Agullo-Lopez, Opt. Commun., 1995, 114: 161

[49] T. Alieva and F. Agullo-Lopez, Opt. Commun., 1996, 115: 267

[50] D. F. V. James and G. S. Agarwal, Opt. Commun., 1996, 126: 207

[51] H. Weyl, Z. Phys., 1927, 46: 1

[52] H. Y. Fan and T. Q. Song, J. Phys. A, 2003, 36: 7803

[53] H. Y. Fan, Phys. Rev. A, 2002, 65: 064102

[54] H. Y. Fan, Phys. Lett. A, 2002, 294: 253

[55] H. Y. Fan and Y. Fan, Phys. Rev. A, 1996, 54: 958

[56] R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A, 1989, 50: 5274

[57] C. Silberhorn, P. K. Lam, O.Weiss, F. König, N. Korolkova, and G. Leuchs, Phys. Rev. Lett., 2001, 86: 4267

[58] E. C. G. Sudarshan, Phys. Rev. Lett., 1963, 10: 277

[59] H. Y. Fan and B. Z. Chen, Phys. Rev. A, 1996, 53: 1948

[60] H. J. Wu and H. Y. Fan, Mod. Phys. Lett. B, 1997, 11: 544

[61] H. Y. Fan, H. Zou, and Y. Fan, Phys. Lett. A, 1999, 254: 137

[62] D. Casasent and D. Psaltis, Proc. IEEE, 1977, 65: 77

[63] H. Y. Fan, Opt. Commun., 2008, 281: 2023

[64] A. Erdelyi, Higher Transcendental Functions, The Bateman Manuscript Project, New York: McGraw-Hill, 1953

[65] H. Y. Fan and H. L. Lu, Phys. Lett. A, 2005, 334: 132

[66] A. Gerrard and N. M. Burch, Introduction to Matrix Methods in Optics, London: John Wiley & Sons, 1975

[67] H. Y. Fan, Commun. Theor. Phys., 2002, 38: 147

[68] H. Y. Fan and J. H. Chen, Commun. Theor. Phys., 2003, 40: 7

[69] H. Y. Fan and Z.-H. Xu, Phys. Rev. A, 1994, 50: 2921

[70] H. Y. Fan and J. VanderLinde, Phys. Rev. A, 1989, 39: 2987

[71] H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics, Shanghai: Shanghai Scientific and Technical Publishers, 1997 (in Chinese)

[72] H. Y. Fan, Entangled State Representations in Quantum Mechanics and Their Applications, Shanghai: Shanghai Jiao Tong University Press, 2001310

[73] H. Kogelnik and T. Li, Appl. Opt., 1966, 5: 1550

[74] P. A. B′elanger, Opt. Lett., 1991, 16: 196

[75] V. Magni, G. Cerullo, and S. D. Silvestri, Opt. Commun., 1993, 96: 348

[76] H. Y. Fan and L. Y. Hu, Opt. Commun., 2008, 281: 1629

[77] M. Nazarathy and J. Shamir, J. Opt. Soc. Am., 1982, 72: 398

[78] M. Nazarathy and J. Shamir, J. Opt. Soc. Am., 1982, 72: 356

[79] M. Nazarathy, A. Hardy, and J. Shamir, J. Opt. Soc. Am. A, 1986, 3: 360

[80] H. Y. Fan and A. Wünsche, Commun. Theor. Phys., 2003, 39: 717

[81] K. Vogel and H. Risken, Phys. Rev. A, 1989, 40: 2847

[82] W. Vogel and W. P. Schleich, Phys. Rev. A, 1991, 44: 7642

[83] U. Leonhardt, Measuring the Quantum State of Light, Cambridge: Cambridge University Press, 1997, and references therein

[84] M. Aspelmeyer, Nat. Phys., 2009, 5: 11

[85] I. L. Chuang and M. A. Nielson, Mod. Opt., 1997, 44: 2455

[86] W. P. Schleich, Quantumm Optics in Phase Space, Berlin: Wiley-VCH, 2001

[87] A. Wünsche, J. Mod. Opt., 1997, 44: 2293

[88] A. Wünsche, Phys. Rev. A, 1996, 54: 5291

[89] H. Y. Fan and L. Y. Hu, Opt. Commun., 2009, 282: 3734

[90] R. F. O’Connell and E. P. Wigner, Phys. Lett. A, 1981, 83: 145

[91] G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2161

[92] G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2187

[93] G. S. Agawal and E. Wolf, Phys. Rev. D, 1972, 2: 2206

[94] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep., 1984, 106: 121

[95] A. C. McBride, and F. H. Kerr, IMA J. Appl. Math., 1987, 39: 159

[96] Y. G. Cai and S. Y. Zhu, Opt. Lett., 2005, 30: 388

[97] S. Liu, J. Xu, Y. Zhang, L. Chen, and C. Li, Opt. Lett., 1995, 20: 1053

[98] C. C. Shih, Opt. Lett., 1995, 20: 1178

[99] L. M. Bernardo and O. D. Soares, Appl. Opt., 1996, 35: 3163

[100] L. M. Bernardo and O. D. Soares, J. Opt. Soc. Am. A, 1994, 11: 2622

[101] S. Chountasis, A. Vourdas, and C. Bendjaballah, Phys. Rev. A, 1999, 60: 3467

[102] H. Y. Fan and L. Y. Hu, Chin. Phys. Lett., 2008, 25: 513

[103] H. M. Ozaktas and M. F. Erden, Opt. Commun., 1997, 143: 75

[104] H. Y. Fan and L. Y. Hu, J. Mod. Opt., 2009, 56: 1819

[105] H. Lee, Phys. Rep., 1995, 259: 147

[106] H. Y. Fan, Opt. Lett., 2003, 28: 2177

[107] H. Y. Fan and H. L. Lu, Opt. Lett., 2006, 31: 2622

[108] L. Y. Hu and H. Y. Fan, J. Mod. Opt., 2008, 55: 2429

[109] H. Y. Fan, L. Y. Hu, and J. S. Wang, J. Opt. Soc. Am. A, 2009, 25: 974

[110] H. Y. Fan and C. H. Lv, J. Opt. Soc. Am. A, 2009, 26: 2306

[111] D. Dragoman, J. Opt. Soc. Am. A, 2009, 26: 274

[112] H. Y. Fan, Commun. Theor. Phys., 1999, 31: 285

[113] H. Y. Fan and Y. Fan, Commun. Theor. Phys., 2000, 33: 701

[114] H. Y. Fan and L. S. Li, Commun. Theor. Phys., 1998, 29: 477

[115] P. Pellat-Finet, Opt. Lett., 1994, 19: 1388

[116] See: e.g., S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology, SIAM, Philadelphia, 2001

[117] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Book/Cole, 2002

[118] C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction toWavelets andWavelet Transforms: A Primer, New Jersey: Prentice Hall, 1998

[119] C. K. Chiu, Introduction to Wavelets, San Diego: Academic Press, 1992

[120] D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett., 1978, 12: 33

[121] E. Kapon, J. Matz, and A. Yariv, Opt. Lett., 1984, 10: 125

[122] M. Oka, H. Masuda, Y. Kaneda, and W. Streifer, IEEE J. Quantum Electron., 1992, 28: 1142

[123] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 1992

[124] L. Y. Hu and H. Y. Fan, J. Mod. Opt., 2008, 55: 1835

[125] P. S. Addison, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance, London: Institute of Physics Publishing, 2002

[126] P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A, 1995, 51: R1750

[127] S. De Luca and E. Fiordilino, J. Phys. B, 1996, 29: 3277

[128] K. Husimi, Proc. Phys.-Math. Soc. Jpn., 1940, 22: 264

[129] C. L. Mehta, Phys. Rev. Lett., 1967, 18: 752

[130] H. Y. Fan, Ann. Phys., 2007, doi: 10.1016/j.aop.2007.06.003

[131] L. Y. Hu and H. Y. Fan, Int. J. Theor. Phys., 2009, 48: 1539

[132] H. Y. Fan and H. L. Lu, Opt. Lett., 2006, 31: 3432

[133] H. Y. Fan and S. G. Liu, Opt. Lett., 2007, 32: 1507

[134] H. Y. Fan, S. G. Liu, and L. Y. Hu, Opt. Lett., 2009, 34: 551

[135] H. Y. Fan and H. L. Lu, J. Phys. A, 2004, 37: 10993

[136] H. Y. Fan, X. B. Tan, and H. L. Lu, Phys. Lett. A, 2006, 357: 163

[137] M. A. Nielsen and I. L. Chuang, The Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000

[138] See: e.g., J. Preskill, Quantum Information and Computation, California Institute of Technology, 1998

[139] S. Parker, S. Bose, and M. Plenio, Phys. Rev. A, 2000, 61: 032305

[140] C. M. Xie and H. Y. Fan, J. Mod. Opt., 2010, 57: 582

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg