Skip to main content
SpringerLink
Log in

Multi-variable special polynomials using an operator ordering method

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. E. Hoffmann, V. Hussin, I. Marquette, and Y. Z. Zhang, Nonclassical behaviour of coherent states for systems constructed using exceptional orthogonal polynomials, J. Phys. A Math. Theor. 51(8), 085202 (2018)

    MATH  Google Scholar 

  2. K. W. Hwang and C. S. Ryoo, Differential equations associated with two variable degenerate Hermite polynomial, Mathematics 8(2), 228 (2020)

    Google Scholar 

  3. H. Y. Fan and Y. Fan, New eigenmodes of propagation in quadratic graded index media and complex fractional fourier transform, Commum. Theor. Phys. 39(1), 97 (2003)

    ADS  MathSciNet  Google Scholar 

  4. H. Y. Fan and J. R. Klauder, Weyl correspondence and Prepresentation as operator Fredholm equations and their solutions, J. Phys. Math. Gen. 39(34), 10849 (2006)

    ADS  MATH  Google Scholar 

  5. H. Y. Fan and J. Chen, Atomic coherent states studied by virtue of the EPR entangled state and their Wigner functions, Eur. Phys. J. D 23(3), 437 (2003)

    ADS  Google Scholar 

  6. X. G. Meng, J. S. Wang, and H. Y. Fan, Atomic coherent state as the eigenstates of a two-dimensional anisotropic harmonic oscillator in a uniform magnetic field, Mod. Phys. Lett. A 24(38), 3129 (2009)

    ADS  MATH  Google Scholar 

  7. H. Y. Fan, Z. L. Wan, Z. Wu, and P. F. Zhang, A new kind of special function and its application, Chin. Phys. B 24(10), 100302 (2015)

    ADS  Google Scholar 

  8. J. S. Wang, X. G. Meng, and H. Y. Fan, Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem, Chin. Phys. B 28(10), 100301 (2019)

    ADS  Google Scholar 

  9. X. G. Meng, J. M. Liu, J. S. Wang, and H. Y. Fan, New generalized binomial theorems involving two-variable Hermite polynomials via quantum optics approach and their applications, Eur. Phys. J. D 73(2), 32 (2019)

    ADS  Google Scholar 

  10. H. Y. Fan and T. F. Jiang, Two-variable Hermite polynomials as time-evolutional transition amplitude for driven harmonic oscillator, Mod. Phys. Lett. B 21(08), 475 (2007)

    ADS  MATH  Google Scholar 

  11. H. Y. Fan and Y. Fan, New eigenmodes of propagation in quadratic graded index media and complex fractional Fourier transform, Commun. Theor. Phys. 39(1), 97 (2003)

    ADS  MathSciNet  Google Scholar 

  12. H. Y. Fan and X. F. Xu, Talbot effect in a quadraticindex medium studied with two-variable Hermite polynomials and entangled states, Opt. Lett. 29(10), 1048 (2004)

    ADS  Google Scholar 

  13. V. V. Dodonov, Asymptotic formulae for two-variable Hermite polynomials, J. Phys. Math. Gen. 27(18), 6191 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  14. H. Y. Fan and Y. Fan, New bosonic operator ordering identities gained by the entangled state representation and two-variable Hermite polynomials, Commum. Theor. Phys. 38(3), 297 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  15. X. G. Meng, J. S. Wang, Z. S. Yang, X. Y. Zhang, Z. T. Zhang, B. L. Liang, and K. C. Li, Squeezed Hermite polynomial state: Nonclassical features and decoherence behavior, J. Opt. 22(1), 015201 (2020)

    ADS  Google Scholar 

  16. H. Y. Fan and X. Ye, Hermite polynomial states in two-mode Fock space, Phys. Lett. A 175(6), 387 (1993)

    ADS  MathSciNet  Google Scholar 

  17. J. A. Bergou, M. Hillery, and D. Q. Yu, Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states, Phys. Rev. A 43(1), 515 (1991)

    ADS  Google Scholar 

  18. H. L. Zhang, H. C. Yuan, L. Y. Hu, and X. X. Xu, Synthesis of Hermite polynomial excited squeezed vacuum states from two separate single-mode squeezed vacuum states, Opt. Commun. 356(23), 223 (2015)

    ADS  Google Scholar 

  19. K. Górska, A. Horzela, and F. H. Szafraniec, Holomorphic Hermite polynomials in two variables, J. Math. Anal. Appl. 470(2), 750 (2019)

    MathSciNet  MATH  Google Scholar 

  20. W. R. Casper, S. Kolb, and M. Yakimov, Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations, arXiv: 2002.07895 (2020)

  21. H. Y. Fan, H. L. Lu, and Y. Fan, Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations, Ann. Phys. 321(2), 480 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  22. H. Y. Fan, Entanged states, squeezed states gained via the route of developing Dirac’s sybmolic method and their applications, Int. J. Mod. Phys. B 18(10n11), 1387 (2004)

    ADS  MATH  Google Scholar 

  23. X. G. Meng, J. S. Wang, B. L. Liang, and C. X. Han, Evolution of a two-mode squeezed vacuum for amplitude decay via continuous-variable entangled state approach, Front. Phys. 13(5), 130322 (2018)

    Google Scholar 

  24. X. G. Meng, Z. Wang, H. Y. Fan, and J. S. Wang, Nonclassicality and decoherence of photon-subtracted squeezed vacuum states, J. Opt. Soc. Am. B 29(11), 3141 (2012)

    ADS  Google Scholar 

  25. X. G. Meng, J. S. Wang, B. L. Liang, and C. X. Du, Optical tomograms of multiple-photon-added Gaussian states via the intermediate state representation theory, J. Exp. Theor. Phys. 127(3), 383 (2018)

    ADS  Google Scholar 

  26. H. Y. Fan and J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum, Phys. Rev. A 49(2), 704 (1994)

    ADS  MATH  Google Scholar 

  27. H. Y. Fan, H. R. Zaidi, and J. R. Klauder, New approach for calculating the normally ordered form of squeeze operators, Phys. Rev. D 35(6), 1831 (1987)

    ADS  MathSciNet  Google Scholar 

  28. J. S. Wang, X. G. Meng, and H. Y. Fan, s-parameterized Weyl transformation and the corresponding quantization scheme, Chin. Phys. B 24(1), 014203 (2015)

    ADS  Google Scholar 

  29. C. X. Du, X. G. Meng, R. Zhang, and J. S. Wang, Analytical and numerical investigations of displaced thermal state evolutions in a laser process, Chin. Phys. B 26(12), 120301 (2017)

    ADS  Google Scholar 

  30. J. H. Liu, Y. B. Zhang, Y. F. Yu, and Z. M. Zhang, Photon-phonon squeezing and entanglement in a cavity optomechanical system with a flying atom, Front. Phys. 14(1), 12601 (2019)

    ADS  Google Scholar 

  31. C. J. Liu, W. Ye, W. D. Zhou, H. L. Zhang, J. H. Huang, and L. Y. Hu, Entanglement of coherent superposition of photon-subtraction squeezed vacuum, Front. Phys. 12(5), 120307 (2017)

    ADS  Google Scholar 

  32. V. V. Dodonov, “Nonclassical” states in quantum optics: A “squeezed” review of the first 75 years, J. Opt. B: Quantum Semiclass. Opt. 4(1), R1 (2002)

    ADS  MathSciNet  Google Scholar 

  33. J. Wenger, R. Tualle-Brouri, and P. Grangier, Non-Gaussian statistics from individual pulses of squeezed light, Phys. Rev. Lett. 92(15), 153601 (2004)

    ADS  Google Scholar 

  34. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Photon subtracted squeezed states generated with periodically poled KTiOPO4, Opt. Express 15(6), 3568 (2007)

    ADS  Google Scholar 

  35. H. Y. Fan, New antinormal ordering expansion for density operators, Phys. Lett. A 161(1), 1 (1991)

    ADS  MathSciNet  Google Scholar 

  36. J. R. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131(6), 2766 (1963)

    ADS  MathSciNet  MATH  Google Scholar 

  37. L. Y. Hu and H. Y. Fan, Statistical properties of photon-subtracted squeezed vacuum in thermal environment, J. Opt. Soc. Am. B 25(12), 1955 (2008)

    ADS  MathSciNet  Google Scholar 

  38. H. Y. Fan, X. G. Meng, and J. S. Wang, New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics, Commum. Theor. Phys. 46(5), 845 (2006)

    ADS  Google Scholar 

  39. S. Y. Lee and H. Nha, Quantum state engineering by a coherent superposition of photon subtraction and addition, Phys. Rev. 82(5), 053812 (2010)

    Google Scholar 

  40. Z. R. Zhong, X. Wang, and W. Qin, Towards quantum entanglement of micromirrors via a two-level atom and radiation pressure, Front. Phys. 13(5), 130319 (2018)

    Google Scholar 

  41. T. Liu, Z. F. Zheng, Y. Zhang, Y. L. Fang, and C. P. Yang, Transferring entangled states of photonic cat-state qubits in circuit QED, Front. Phys. 15(2), 21603 (2020)

    ADS  Google Scholar 

  42. H. C. Yuan, X. X. Xu, and Y. J. Xu, Generating two variable Hermite polynomial excited squeezed vacuum states by conditional measurement on beam splitters, Optik 172(21), 1034 (2018)

    ADS  Google Scholar 

  43. P. L. Kelley and W. H. Kleiner, Theory of electromagnetic field measurement and photoelectron counting, Phys. Rev. 136(2A), A316 (1964)

    ADS  MathSciNet  MATH  Google Scholar 

  44. H. Y. Fan and L. Y. Hu, Two quantum-mechanical photocount formulas, Opt. Lett. 33(5), 443 (2008)

    ADS  Google Scholar 

  45. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge: Cambridge University Press, 1997

    Google Scholar 

Download references

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No. 11347026) and the Natural Science Foundation of Shandong Province (Grant Nos. ZR2016AM03 and ZR2017MA011).

Author information

Authors and Affiliations

  1. Shandong Provincial Key Laboratory of Optical Communication Science and Technology, School of Physical Science and Information Engineering, Liaocheng University, Liaocheng, 252059, China

    Xiang-Guo Meng, Zhen-Shan Yang, Xiao-Yan Zhang, Zhen-Tao Zhang & Bao-Long Liang

  2. School of Physics and Electronic Engineering, Linyi University, Linyi, 276000, China

    Kai-Cai Li

  3. Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, College of Physics and Engineering, Qufu Normal University, Qufu, 273165, China

    Kai-Cai Li & Ji-Suo Wang

Authors
  1. Xiang-Guo Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai-Cai Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ji-Suo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen-Shan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiao-Yan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zhen-Tao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Bao-Long Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Xiang-Guo Meng, Kai-Cai Li or Ji-Suo Wang.

Additional information

Author contribution statement

All the authors contributed in writing and revising the manuscript. Xiang-Guo Meng and Kai-Cai Li contributed equally to this work and should be considered as co-first authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, XG., Li, KC., Wang, JS. et al. Multi-variable special polynomials using an operator ordering method. Front. Phys. 15, 52501 (2020). https://doi.org/10.1007/s11467-020-0967-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-020-0967-3

Keywords

Search

Navigation