Skip to main content

Advertisement

Log in

Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξqD(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e.,

$ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $

and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξq is stronger for odd q than for even q.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Schleich, W.P.: Quantum Optics in Phase Space. Wiley, Hoboken (2001)

    Book  MATH  Google Scholar 

  2. Carmichael, H.J.: Statistical Methods in Quantum Optics 2: Non-classical Fields. Springer, Berlin (2007)

    Google Scholar 

  3. Zhang, R., Meng, X.G., Du, C.X., Wang, J.S.: Nonclassicality of Photon-Added displaced thermal state via quantum Phase-Space distributions. J. Phys. Soc. Jpn. 87, 024001 (2018)

    Article  ADS  Google Scholar 

  4. Meng, X.G., Sheng, G.H., Wang, J.S., Zhang, R.: Nonclassical thermal-state superpositions: Analytical evolution law and decoherence behavior. Opt. Commun. 411(7), 15–20 (2018)

    Article  ADS  Google Scholar 

  5. Dodonov, V.V., Man’ko, I.: Theory of Nonclassical States of Light. Taylor & Francis, Milton Park (2003)

    Google Scholar 

  6. Meng, X.G., Wang, J.S., Liang, B.L.: New approach for deriving the exact time-evolution of density operator for diffusive anharmonic oscillator and its Wigner distribution function. Chinese Phys. B 22(3), 030307/1–6 (2013)

    Article  ADS  Google Scholar 

  7. Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: fundamentals. Phys. Rep. 106, 121–167 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  8. Biswas, A., Agarwal, G.S.: Nonclassicality and decoherence of photon-subtracted squeezed states. Phys. Rev. A 75, 032104 (2007)

    Article  ADS  Google Scholar 

  9. Hu, L.Y., Fan, H.Y.: Statistical properties of photon-subtracted squeezed vacuum in thermal environment. J. Opt. Soc. Am. B 25, 1955–1964 (2008)

    Article  ADS  Google Scholar 

  10. Meng, X.G., Wang, Z., Wang, J.S., Fan, H.Y.: Nonclassicality and decoherence of photon-subtracted squeezed vacuum states. J. Opt. Soc. Am. B 29, 3141–3149 (2012)

    Article  ADS  Google Scholar 

  11. Wang, Z., Meng, X.G., Fan, H.Y.: Photon-subtracted squeezed coherent state: Nonclassicality and decoherence in thermal environment. J. Opt. Soc. Am. B 29(3), 397–406 (2012)

    Article  ADS  Google Scholar 

  12. Fan, H.Y., Lu, H.L., Fan, Y.: Newton-leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations. Ann. Phys. 321, 480–494 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Meng, X.G., Wang, Z., Wang, J.S., Fan, H.Y.: Wigner function, optical tomography of two-variable Hermite polynomial state, and its decoherence effects studied by the entangled state representations. J. Opt. Soc. Am. B 30(6), 1614–1622 (2013)

    Article  ADS  Google Scholar 

  14. Meng, X.G., Wang, Z., Fan, H.Y., Wang, J.S., Yang, Z.S.: Nonclassical properties of photon-added two-mode squeezed thermal state and their decoherence in the thermal channel. J. Opt. Soc. Am. B 29(7), 1844–1853 (2012)

    Article  ADS  Google Scholar 

  15. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  16. Cochrane, P.T., Milburn, G.J.: Teleportation with the entangled states of a beam splitter. Phys. Rev. A 64(06), 2001 (2312)

    Google Scholar 

  17. Wang, J.S., Fan, H.Y., Meng, X.G.: A generalized Weyl-Wigner quantization scheme unifying P-Q and Q-P ordering and Weyl ordering of operators. Chin. Phys. B 21, 064204 (2012)

    Article  ADS  Google Scholar 

  18. Wang, J.S., Meng, X.G., Liang, B.L.: Wave function for the squeezed atomic coherent state in entangled state representation and some of its applications. Chin. Phys. B 19, 014207 (2010)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  20. Fan, H.Y., Chen, J.H., Zhang, P.F.: On the entangled fractional squeezing transformation. Front. Phys. 10, 187–191 (2015)

    Article  Google Scholar 

  21. Vaidman, L.: Teleportation of quantum states. Phys. Rev. A 49, 1473–1476 (1994)

    Article  ADS  Google Scholar 

  22. Braunstein, S.L., Kimble, H.J.: Teleportation of continuous quantum variables. Phys. Rev. Lett. 80, 869–872 (1998)

    Article  ADS  Google Scholar 

  23. Fan, H.Y., Zaidi, H.R.: Application of IWOP technique to the generalized Weyl correspondence. Phys. Lett. A 124, 303–307 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  24. Meng, X.G., Wang, J.S., Liang, B.L., Du, C.X.: Optical tomograms of multiple-photon-added Gaussian states via the intermediate state representation theory. J. Exp. Theor. Phys. 127, 383–390 (2018)

    Article  ADS  Google Scholar 

  25. Kenfack, A., Życzkowski, K.: Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B: Quantum Semiclass. Opt. 6, 396–404 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  26. Fan, H.Y.: Entangled State Representations in Quantum Mechanics and Their Applications. Shanghai Jiao Tong Univ. Press, Shanghai (2001)

    Google Scholar 

  27. Fan, H.Y., Lou, S.Y., Pan, X.Y., Da, C.: Binomial theorem involving Hermite polynomials and negative-binomial theorem involving Laguerre polynomials. Acta. Phys. Sin. 62, 240301 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ji-Suo Wang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, KC., Meng, XG. & Wang, JS. Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter. Int J Theor Phys 58, 2521–2530 (2019). https://doi.org/10.1007/s10773-019-04142-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-019-04142-3

Keywords

Navigation