Lithuanian Journal of Physics article / Lietuvos fizikos žurnalo straipsnis

NEW GENERALIZED HERMITE POLYNOMIALS WITH THREE VARIABLES OBTAINED VIA QUANTUM OPTICS METHOD AND THEIR APPLICATIONS
An-peng Wang^a and Yi-xing Wang^b
^aShandong Huayu University of Technology, 253000 Dezhou, China
^bSchool of Automation and Electrical Engineering, Linyi University, 276000 Linyi, China
Emails: 823674870@qq.com; wangyixing@lyu.edu.cn

Received 13 January 2025; revised 10 May 2025; accepted 20 May 2025

Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.
Keywords: generalized Hermite polynomial, generating function, operator ordering method, normalization, Wigner function

NAUJI APIBENDRINTIEJI ERMITO POLINOMAI SU TRIMIS KINTAMAISIAIS, GAUTI KVANTINĖS OPTIKOS METODU, IR JŲ TAIKYMAI
An-peng Wang^a, Yi-xing Wang^b

^aŠandongo Huayu technologijos universitetas, Dedžou, Kinija
^bLinyi universiteto Automatikos ir elektrotechnikos mokykla, Linyi, Kinija

References / Nuorodos

[1] K.W. Hwang and C.S. Ryoo, Differential equations associated with two variable degenerate Hermite polynomials, Mathematics 8(2), 228/1–18 (2020),
https://doi.org/10.3390/math8020228
[2] 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), 97–100 (2003),
https://doi.org/10.1088/0253-6102/39/1/97
[3] 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),
https://doi.org/10.1007/s11467-018-0856-1
[4] 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),
https://doi.org/10.1088/1674-1056/24/10/100302
[5] 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(8), 475–480 (2007),
https://doi.org/10.1142/S0217984907012943
[6] H.Y. Fan and X.F. Xu, Talbot effect in a quadratic index medium studied with two-variable Hermite polynomials and entangled states, Opt. Lett. 29(10), 1048 (2004),
https://doi.org/10.1364/OL.29.001048
[7] X.G. Meng, H.-S. Goan, J.S. Wang, and R. Zhang, Nonclassical thermal-state superpositions: Analytical evolution law and decoherence behavior, Opt. Commun. 411(7), 15–20 (2018),
https://doi.org/10.1016/j.optcom.2017.11.005
[8] H. Zhang, W. Ye, C.P. Wei, Y. Xia, S.K. Chang, Z.Y. Liao, and L.Y. Hu, Improved phase sensitivity in a quantum optical interferometer based on multiphoton catalytic two-mode squeezed vacuum states, Phys. Rev. A 103(1), 013705 (2021),
https://doi.org/10.1103/PhysRevA.103.013705
[9] S. Wang, X.X. Xu, Y.J. Xu, and L.J. Zhang, Quantum interferometry via a coherent state mixed with a photon-added squeezed vacuum state, Opt. Commun. 444, 102–110 (2019),
https://doi.org/10.1016/j.optcom.2019.03.068
[10] 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), 383–390 (2018),
https://doi.org/10.1134/S1063776118080113
[11] X.G. Meng, B.L. Liang, J.M. Liu, and X.C. Zhou, Entropy evolution law of general quadratic state in a laser channel, Front. Phys. 20, 022205 (2025),
https://doi.org/10.15302/frontphys.2025.022205
[12] X.G. Meng, K.C. Li, J.S. Wang, Z.S. Yang, X.Y. Zhang, Z.T. Zhang, and B.L. Liang, Multivariable special polynomials using an operator ordering method, Front. Phys. 15(5), 52501 (2020),
https://doi.org/10.1007/s11467-020-0967-3
[13] S.Y. Lee and H. Nha, Quantum state engineering by a coherent superposition of photon subtraction and addition, Phys. Rev. 82(5), 053812 (2010),
https://doi.org/10.1103/PhysRevA.82.053812
[14] X.G. Meng, J.S. Wang, X.Y. Zhang, Z.S. Yang, B.L. Liang, and Z.T. Zhang, Nonclassicality via the superpositions of photon addition and subtraction and quantum decoherence for thermal noise, Ann. Phys. (Berlin) 532(10), 2000219 (2020),
https://doi.org/10.1002/andp.202000219
[15] S. Wang, L.L. Hou, X.F. Chen, and X.F. Xu, Continuous-variable quantum teleportation with non-Gaussian entangled states generated via multiple-photon subtraction and addition, Phys. Rev. A 91(6), 063832 (2015),
https://doi.org/10.1103/PhysRevA.91.063832
[16] X.G. Meng, K.C. Li, J.S. Wang, X.Y. Zhang, Z.T. Zhang, Z.S. Yang, and B.L. Liang, Continuous-variable entanglement and Wigner-function negativity via adding or subtracting photons, Ann. Phys. (Berlin) 532(5), 1900585 (2020),
https://doi.org/10.1002/andp.201900585
[17] H.Y. Fan and L.Y. Hu, Operator-sum representation of density operators as solutions to master equations obtained via the entangled state approach, Mod. Phys. Lett. B 22(25), 2435–2468 (2008),
https://doi.org/10.1142/S0217984908017072
[18] X.G. Meng, Z. Wang, J.S. Wang, and H.Y. Fan, 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),
https://doi.org/10.1364/JOSAB.30.001614