Skip to main content
SpringerLink
Log in

Hong and Mandel fourth-order squeezing generated by the beam splitter with third-order nonlinearity from the coherent light

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

We study about the generation of Hong and Mandel fourth-order squeezing by the beam splitter with the third-order nonlinearity. We find that when we mix coherent light beam (a state which lies at the boundary of classical to non-classical region), we find situations where the output beam exhibit Hong and Mandel fourth-order squeezing.

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

Fig. 1

Similar content being viewed by others

References

  • Abbott, B.P., et al.: GW150914: the advanced LIGO detectors in the era of first discoveries. Phys. Rev. Lett. 116, 131103 (2016)

    ADS  MathSciNet  Google Scholar 

  • Becker, W., Scully, M.O., Zubairy, M.S.: Generation of squeezed coherent states via a free-electron laser. Phys. Rev. Lett. 48, 475–477 (1982)

    ADS  Google Scholar 

  • Breitenbach, G., Mueller, T., Pereira, S.F., Poizat, J-Ph, Schiller, S., Mlynek, J.: Squeezed vacuum from a monolithic optical parametric oscillator. J. Opt. Soc. Am. B 12, 2304–2309 (1995)

    ADS  Google Scholar 

  • Cahill, K.E., Glauber, R.J.: Density operators and quasiprobability distributions. Phys. Rev. 177, 1882–1902 (1969)

    ADS  Google Scholar 

  • Campos, R.A., Saleh, B.E.A., Teich, M.C.: Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics. Phys. Rev. A 40, 1371–1384 (1989)

    ADS  Google Scholar 

  • Caron, B., et al.: The Virgo interferometer. Class. Quantum Grav. 14, 1461–1469 (1997)

    ADS  Google Scholar 

  • Chandra, N., Prakash, H.: Anticorrelation in two-photon attenuated laser beam. Phys. Rev. A 1, 1696–1698 (1970)

    ADS  Google Scholar 

  • Chandra, N., Prakash, H.: Regularized P-representation. Phys. Rev. D 7, 1936–1937 (1973)

    ADS  Google Scholar 

  • Dakna, M., Anhut, T., Opatrny, T., Knoll, L., Welsch, D.-G.: Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter. Phys. Rev. A 55, 3184–3194 (1997)

    ADS  Google Scholar 

  • Dodonov, V.V., Man’ko, V.I. (eds.): Theory of Nonclassical States of Light. Taylor and Francis, London (2003)

    Google Scholar 

  • Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963a)

    ADS  MathSciNet  MATH  Google Scholar 

  • Glauber, R.J.: Photon correlations. Phys. Rev. Lett. 10, 84–86 (1963b)

    ADS  MathSciNet  Google Scholar 

  • Glauber, R.J.: The quantum theory of optical coherence. Phys. Rev. 130, 2529–2539 (1963c)

    ADS  MathSciNet  Google Scholar 

  • Gong, J.J., Aravind, P.K.: Higher-order squeezing in three- and four-wave mixing processes with loss. Phys. Rev. A 46, 1586–1593 (1992)

    ADS  Google Scholar 

  • Hello, P.: II: optical aspects of interferometric gravitational-wave detectors. In: Wolf, E. (ed.) Progress in Optics XXXVII, p. 85. Elsevier, Amsterdam (1998)

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  • Hong, C.K., Mandel, L.: Higher-order squeezing of a quantum field. Phys. Rev. Lett. 54, 323–325 (1985a)

    ADS  Google Scholar 

  • Hong, C.K., Mandel, L.: Generation of higher-order squeezing of quantum electromagnetic fields. Phys. Rev. A 32, 974–982 (1985b)

    ADS  Google Scholar 

  • Huttner, B., Ben-Aryeh, Y.: Interference of a beam splitter on photon statistics. Phys. Rev. A 38, 204–211 (1988)

    ADS  Google Scholar 

  • Janszky, J., Sibilia, C., Bertolotti, M., Yushin, Y.: Non-classical light in a linear coupler. J. Mod. Opt. 35, 1757–1765 (1988)

    ADS  Google Scholar 

  • Janszky, J., Sibilia, C., Bertolotti, M.: Photon number-distribution in a linear directional coupler. J. Mod. Opt. 38, 2467–2477 (1991a)

    ADS  Google Scholar 

  • Janszky, J., Adam, P., Sibilia, C., Bertolotti, M.: Amplitude-squeezed number-phase intelligent states in a directional coupler. Quantum Opt. 4, 163–172 (1991b)

    ADS  Google Scholar 

  • Kimble, H.J., Dagenais, M., Mandel, L.: Photon antibunching in resonance fluorescence. Phys. Rev. Lett. 39, 691–695 (1977)

    ADS  Google Scholar 

  • Klauder, J.R.: Improved version of optical equivalence theorem. Phys. Rev. Lett. 16, 534–536 (1966)

    ADS  MathSciNet  Google Scholar 

  • Klauder, J.R., Sudarshan, E.C.G.: Fundamentals of Quantum Optics. W. A. Benjamin, New York (1968)

    Google Scholar 

  • Klauder, J.R., McKenna, J., Currie, D.G.: On “diagonal” coherent-state representations for quantum-mechanical density matrices. J. Math. Phys. 6, 733–739 (1965a)

    ADS  MathSciNet  MATH  Google Scholar 

  • Klauder, J.R., McKenna, J., Currie, D.G.: On “Diagonal” Coherent-state representations for quantum-mechanical density matrices. J. Math. Phys. 6, 734–739 (1965b)

    ADS  MathSciNet  MATH  Google Scholar 

  • Klien, F.L., Kozierowski, M., Quang, T.: Fourth-order squeezing in the multiphoton Jaynes–Cummings model. Phys. Rev. A 38, 263–266 (1988)

    ADS  Google Scholar 

  • Kozierowski, M.: Higher-order squeezing in kth-harmonic generation. Phys. Rev. A 34, 3474–3477 (1986)

    ADS  Google Scholar 

  • Kozierowski, M., Kielich, S.: Squeezed states in harmonic generation of laser beam. Phys. Lett. A 94, 213–216 (1983)

    ADS  Google Scholar 

  • Louisell, W.H.: Quantum Statistical Properties of Radiation. Wiley, New York (1973)

    MATH  Google Scholar 

  • Lvovsky, A.I.: Squeezed light. In: Andrews, D. (ed.) Photonics Volume 1: Fundamentals of Photonics and Physics, pp. 121–164. Wiley, West Sussex (2015)

    Google Scholar 

  • Lvovsky, A.I., Mlynek, J.: Quantum-optical catalysis: generating nonclassical states of light by means of linear optics. Phys. Rev. Lett. 88, 250401–250404 (2002)

    ADS  Google Scholar 

  • Mandel, L.: Squeezing and photon antibunching in harmonic generation. Opt. Commun. 42, 437–439 (1982)

    ADS  Google Scholar 

  • Mehta, C.L., Sudarshan, E.C.G.: Relation between quantum and semiclassical description of optical coherence. Phys. Rev. 138, B274–B280 (1965)

    ADS  MathSciNet  Google Scholar 

  • Milburn, G.J., Walls, D.F.: Production of squeezed states in a degenerate parametric amplifier. Opt. Comm. 39, 401–404 (1981)

    ADS  Google Scholar 

  • Miller, M.M., Mishkin, E.A.: Characteristic states of the electromagnetic radiation field. Phys. Rev. 152, 1110–1114 (1966)

    ADS  MATH  Google Scholar 

  • Paul, H.: Photon antibunching. Rev. Mod. Phys. 54, 1061–1102 (1982)

    ADS  Google Scholar 

  • Prakash, H., Mishra, D.K.: An example of enhancement of a non-classical feature of a light beam by mixing with another classical light beam using a beam splitter. J. Phys. B At. Mol. Opt. Phys. 38, 665–670 (2005)

    ADS  Google Scholar 

  • Prakash, H., Mishra, D.K.: Influence of a symmetric lossless beam splitter on a non-classical feature of an optical field. Opt. Spectrosc. 103(1), 145–147 (2007a)

    ADS  MathSciNet  Google Scholar 

  • Prakash, H., Mishra, D.K.: Enhancement and generation of sum squeezing in two-mode light in mixing with coherent light using a beam splitter. Eur. Phys. J. D 45(2), 363–367 (2007b)

    ADS  Google Scholar 

  • Prakash, H., Mishra, D.K.: Generation of nonclassical optical fields by a beam splitter with third-order nonlinearity. Opt. Lett. 35(13), 2212–2214 (2010)

    ADS  Google Scholar 

  • Prakash, H., Mishra, D.K.: Quantum analysis of a beam splitter with second-order nonlinearity and generation of nonclassical light. J. Opt. Soc. Am. B 33(7), 1552–1557 (2016)

    ADS  Google Scholar 

  • Reid, M.D., Walls, D.F.: Generation of squeezed states via degenerate four-wave mixing. Phys. Rev. A 31, 1622–1635 (1985)

    ADS  Google Scholar 

  • Glauber, R.J.: Optical coherence and photon statistics. In: de Witt, C., Blandin, A., Cohen-Tannoudji C. (eds.) Quantum Optics and Electronics. Gordon and Breach, New York (1965)

    Google Scholar 

  • Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dušek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009)

    ADS  Google Scholar 

  • Schnabel, R.: Squeezed states of light and their applications in laser interferometers. Phys. Rep. 684, 1–51 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  • Shelby, R.M., Levenson, M.D., Perlmutter, S.H., Devoe, R.G., Walls, D.F.: Broad-band parametric deamplification of quantum noise in an optical fiber. Phys. Rev. Lett. 57, 691–694 (1986)

    ADS  Google Scholar 

  • Short, R., Mandel, L.: Observation of sub-Poissonian photon statistics. Phys. Rev. Lett. 51, 384–387 (1983)

    ADS  Google Scholar 

  • Slusher, R.E., Hollberg, L.W., Yurke, B., Mertz, J.C., Valley, J.F.: Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett. 55, 2409–2412 (1985)

    ADS  Google Scholar 

  • Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963)

    ADS  MathSciNet  MATH  Google Scholar 

  • Walls, D.F.: Squeezed states of light. Nature 306, 141–146 (1983)

    ADS  Google Scholar 

  • Walls, D.F., Zoller, P.: Reduced quantum fluctuations in resonance fluorescence. Phys. Rev. Lett. 47, 709–711 (1981)

    ADS  Google Scholar 

  • Wu, L.A., Kimble, H.J., Hall, J.L., Wu, H.: Generation of squeezed states by parametric down conversion. Phys. Rev. Lett. 57, 2520–2523 (1986)

    ADS  Google Scholar 

  • Wu, L.-A., Kimble, H.J., Hall, J.L., Wu, H.: Generation of squeezed states by parametric down conversion. Phys. Rev. Lett. 57, 2520–2523 (1987)

    ADS  Google Scholar 

  • Xizeng, L., Ying, S.: Generation of higher-order squeezing of quantum electromagnetic fields by degenerate four-wave mixing. Phys. Rev. A 40, 7384–7386 (1989)

    ADS  Google Scholar 

  • Yamamoto, Y., Haus, H.A.: Preparation measurement and information capacity of optical quantum states. Rev. Mod. Phys. 58, 1001–1020 (1986). (Review and references therein)

    ADS  Google Scholar 

  • Yuen, H.P.: Generation, detection, and application of high-intensity photon-number-eigenstate fields. Phys. Rev. Lett. 56, 2176–2179 (1986)

    ADS  Google Scholar 

  • Yuen, H.P., Shapiro, J.H.: Generation and detection of two-photon coherent states in degenerate four-wave mixing. Opt. Lett. 4, 334–336 (1979)

    ADS  Google Scholar 

  • Yurke, B.: Use of cavities in squeezed-state generation. Phys. Rev. A 29, 408–410 (1984)

    ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

DKM acknowledges financial supports from the SERB, New Delhi under EMR Project (EMR/2016/001694) and UGC-CAS.

Author information

Authors and Affiliations

  1. Physics Department, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Devendra Kumar Mishra

  2. Physics Department, Nehru Gram Bharati (Deemed to be University), Kotwa-Jamunipur Dubawal, Allahabad, U. P., India

    Vikram Singh

Authors
  1. Devendra Kumar Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vikram Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Devendra Kumar Mishra.

Additional information

Publisher's Note

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

Appendix

Appendix

For simplicity, we take \({\upphi } = 0\), \(\upalpha = \left|\upalpha \right|,\quad\upbeta = \left|\upbeta \right|\), let \(\upalpha = x,\quad\upbeta = y\), x and y are real and positive quantities and we have

$$<{\hat{\text{c}}}^{\dag } {\hat{\text{c}}}> \approx {\text{t}}^{2} {\text{x}}^{2} + {\text{r}}^{2} {\text{y}}^{2} + 2{\text{r}}{\kern 1pt} {\text{t}}({\text{p}} + 2{\text{q}})({\text{x}}^{2} - {\text{y}}^{2} ){\text{xy}}^{3}$$
(20)
$$<{\hat{\text{c}}}> + < {\hat{\text{c}}}^{\dag } > \approx 2{\text{tx}} + 2{\text{p}}({\text{rx}}^{2} {\text{y}} + {\text{ry}}^{3} ) + 2{\text{q}}({\text{rx}}^{2} {\text{y}} - {\text{ry}}^{3} )$$
(21)
$$<{\hat{\text{c}}}^{2}> + < {\hat{\text{c}}}^{\dag 2} > + 2 < {\hat{\text{c}}}^{\dag } {\hat{\text{c}}} > \approx 2{\text{t}}^{2} {\text{x}}^{2} + {\text{p}}(8{\text{rtx}}^{3} {\text{y}} + 8{\text{rtxy}}^{3} + 4{\text{rtxy}}) + {\text{q}}(8{\text{rtx}}^{3} {\text{y}} - 8{\text{rtxy}}^{3} )$$
(22)
$$\begin{aligned} <{\hat{\text{c}}}^{3}> + <{\hat{\text{c}}}^{\dag 3}> & \approx 2{\text{t}}^{3} {\text{x}}^{3} - 6{\text{r}}^{2} {\text{txy}}^{2} + 4{\text{t}}^{2} {\text{p}}({\text{rx}}^{4} {\text{y}} + {\text{rx}}^{2} {\text{y}}^{3} ) + 8{\text{rt}}^{2} {\text{p}}({\text{x}}^{2} {\text{y}}^{3} + {\text{x}}^{2} {\text{y}}) \\ & \quad - 2{\text{rtq}}(2{\text{tx}}^{4} {\text{y}} - 2{\text{tx}}^{2} {\text{y}}^{3} + {\text{tx}}^{2} {\text{y}} - {\text{ty}}^{3} ) \\ & \quad - 2{\text{r}}^{2} {\text{p}}(2{\text{rx}}^{2} {\text{y}}^{3} + 2{\text{rx}}^{2} {\text{y}} + 2{\text{ry}}^{5} + 2{\text{ry}}^{3} + {\text{rx}}^{2} {\text{y}} + {\text{ry}}^{3} ) \\ & \quad - 2{\text{r}}^{2} {\text{q}}(2{\text{rx}}^{2} {\text{y}}^{3} + 2{\text{rx}}^{2} {\text{y}} - 2{\text{ry}}^{5} - 2{\text{ry}}^{3} + {\text{rx}}^{2} {\text{y}} - {\text{ry}}^{3} ) \\ \end{aligned}$$
(23)
$$\begin{aligned} <{\hat{\text{c}}}^{\dag } {\hat{\text{c}}}^{2}> + <{\hat{\text{c}}}^{\dag 2} {\hat{\text{c}}}> & \approx 2{\text{t}}^{3} {\text{x}}^{3} + 2{\text{r}}^{2} {\text{txy}}^{3} + {\text{p}}(4{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} + 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} + 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}} + 4{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} + 4{\text{r}}^{3} {\text{y}}^{5} \\ & \quad + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}} + 2{\text{r}}^{3} {\text{y}}^{3} + 2{\text{rt}}^{2} x^{4} y + 2{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} - 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} - 2{\text{r}}^{3} {\text{y}}^{5} - 8{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} ) \\ & \quad + {\text{q}}(4{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} + 2{\text{rt}}^{2} {\text{x}}^{2} {\text{y}} - 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} - 2{\text{rt}}^{2} {\text{y}}^{3} + 4{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} \\ & \quad - 4{\text{r}}^{3} {\text{y}}^{5} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}} - 2{\text{r}}^{3} {\text{y}}^{3} + 2{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} - 2{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} - 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} + 2{\text{r}}^{3} {\text{y}}^{5} \\ & \quad + 4{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} - 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} ) \\ \end{aligned}$$
(24)
$$\begin{aligned} <{\hat{\text{c}}}^{{^{\dag 2} }} {\hat{\text{c}}}^{2}> & \approx {\text{t}}^{4} {\text{x}}^{4} + {\text{r}}^{4} {\text{y}}^{4} + 2{\text{r}}^{2} {\text{t}}^{2} {\text{x}}^{2} {\text{y}}^{2} + {\text{p}}\left( {4{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} - 4{\text{rt}}^{3} {\text{x}}^{3} {\text{y}}^{3} + 4{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} - 8{\text{r}}^{3} {\text{txy}}^{5} + 2{\text{r}}^{3} {\text{tx}}^{3} {\text{y}} - 2{\text{r}}^{3} {\text{txy}}^{3} } \right) \\ & \quad + {\text{q}}(8{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} + 4{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} - 8{\text{rt}}^{3} {\text{x}}^{3} {\text{y}}^{3} - 4{\text{rt}}^{3} {\text{xy}}^{3} + 8{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}^{3} - 8{\text{r}}^{3} {\text{txy}}^{5} - 4{\text{r}}^{3} {\text{txy}}^{3} + 4{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}) \\ \end{aligned}$$
(25)
$$\begin{aligned} <{\hat{\text{c}}}^{\dag } {\hat{\text{c}}}^{3}> + <{\hat{\text{c}}}^{\dag 3} {\hat{\text{c}}}> & \approx 2{\text{t}}^{4} {\text{x}}^{4} - 2{\text{r}}^{4} {\text{y}}^{4} + {\text{p}}(6{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} + 8{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} - 10{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}^{3} - 6{\text{r}}^{3} {\text{tx}}^{3} {\text{y}} \\ & \quad + 2{\text{r}}^{3} {\text{txy}}^{5} + 6{\text{r}}^{3} {\text{txy}}^{3} + 8{\text{r}}^{2} {\text{t}}^{2} {\text{x}}^{2} {\text{y}}^{4} + 8{\text{r}}^{2} {\text{t}}^{2} {\text{x}}^{2} {\text{y}}^{2} ) \\ & \quad + {\text{q}}(4{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} - 4{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} - 2{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} + 2{\text{rt}}^{3} {\text{xy}}^{3} - 12{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}^{3} \\ & \quad - 2{\text{r}}^{3} {\text{tx}}^{3} {\text{y}} + 10{\text{r}}^{3} {\text{txy}}^{3} + 12{\text{r}}^{3} {\text{txy}}^{5} ) \\ \end{aligned}$$
(26)
$$\begin{aligned} <{\hat{\text{c}}}^{4}> + <{\hat{\text{c}}}^{\dag 4}> & \approx 2{\text{t}}^{4} {\text{x}}^{4} + 2{\text{r}}^{4} {\text{y}}^{4} - 12{\text{r}}^{2} {\text{t}}^{2} {\text{x}}^{2} {\text{y}}^{2} + {\text{p}}(44{\text{rt}}^{3} {\text{x}}^{3} {\text{y}}^{3} + 16{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} + 6{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} \\ & \quad - 14{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}^{3} - 14{\text{r}}^{3} {\text{tx}}^{3} {\text{y}} - 26{\text{r}}^{3} {\text{txy}}^{5} - 34{\text{r}}^{3} {\text{txy}}^{3} + {\text{q}}( - 8{\text{rt}}^{3} {\text{x}}^{5} {\text{y}} - 6{\text{rt}}^{3} {\text{x}}^{3} {\text{y}} \\ & \quad + 8{\text{rt}}^{3} {\text{x}}^{3} {\text{y}}^{3} + 6{\text{rt}}^{3} {\text{xy}}^{3} - 14{\text{r}}^{3} {\text{tx}}^{3} {\text{y}} + 8{\text{r}}^{3} {\text{txy}}^{5} + 14{\text{r}}^{3} {\text{txy}}^{3} - 8{\text{r}}^{3} {\text{tx}}^{3} {\text{y}}^{3} ) \\ \end{aligned}$$
(27)
$$\begin{aligned} <{\hat{\text{c}}}^{\dag } {\hat{\text{c}}}^{2}> + <{\hat{\text{c}}}^{\dag 2} {\hat{\text{c}}}> & \approx 2{\text{t}}^{3} {\text{x}}^{3} + 2{\text{r}}^{2} {\text{txy}}^{3} + {\text{p}}(6{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} - 2{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} + 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}} \\ & \quad + 2{\text{r}}^{3} {\text{y}}^{3} + 2{\text{r}}^{3} {\text{y}}^{5} ) + {\text{q}}(10{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} - 10{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} + 2{\text{rt}}^{2} {\text{x}}^{2} {\text{y}} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}} \\ & \quad - 2{\text{rt}}^{2} {\text{y}}^{3} - 2{\text{r}}^{3} {\text{y}}^{5} ) + \frac{1}{8}{\text{q}}(26{\text{rt}}^{2} {\text{x}}^{4} {\text{y}} - 26{\text{rt}}^{2} {\text{x}}^{2} {\text{y}}^{3} - 4{\text{rt}}^{2} {\text{y}}^{3} + 4{\text{rt}}^{2} {\text{x}}^{2} {\text{y}} + 2{\text{r}}^{3} {\text{x}}^{2} {\text{y}}^{3} - 2{\text{r}}^{3} {\text{y}}^{5} ) \\ \end{aligned}$$
(28)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mishra, D.K., Singh, V. Hong and Mandel fourth-order squeezing generated by the beam splitter with third-order nonlinearity from the coherent light. Opt Quant Electron 52, 68 (2020). https://doi.org/10.1007/s11082-019-2188-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-019-2188-y

Keywords

Search

Navigation