Hermite Coherent States for Quadratic Refractive Index Optical Media

  • Zulema Gress
  • Sara Cruz y CruzEmail author
Part of the CRM Series in Mathematical Physics book series (CRM)


Ladder and shift operators are determined for the set of Hermite–Gaussian modes associated with an optical medium with quadratic refractive index profile. These operators allow to establish irreducible representations of the su(1, 1) and su(2) algebras. Glauber coherent states, as well as su(1, 1) and su(2) generalized coherent states, were constructed as solutions of differential equations admitting separation of variables. The dynamics of these coherent states along the optical axis is also evaluated.


Hermite–Gaussian modes Ladder operators Coherent states Self-focusing media Paraxial beams Ermakov equation 



The financial support of CONACyT, Mexico (Project A1-S-24569 and grant 257292 for ZG), Instituto Politécnico Nacional, Mexico (Project SIP20180377), the Spanish MINECO (Pro. MTM2014-57129-C2-1-P), and Junta de Castilla y León, Spain (VA137G18) is acknowledged. The authors are indebted to Prof. J. Negro for enlightening comments and to the anonymous referee for valuable suggestions. Z. Gress is grateful to the Valladolid University for kind hospitality.


  1. 1.
    G. Nienhuis, L. Allen, Paraxial wave optics and harmonic oscillators. Phys. Rev. A 48, 656 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    S.G. Krivoshlykov, N.I. Petrov, I.N. Sisakyan, Correlated coherent states and propagation of arbitrary Gaussian beams in longitudinally homogeneous quadratic media exhibiting absorption or amplification. Sov. J. Quantum Electron. 16, 933 (1986)ADSCrossRefGoogle Scholar
  3. 3.
    N.I. Petrov, Macroscopic quantum effects for classical light. Phys. Rev. A 90, 043814 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    D. Stoler, Operator methods in physical optics. J. Opt. Soc. Am. 71, 334 (1981)ADSCrossRefGoogle Scholar
  5. 5.
    M.A.M. Marte, S. Stenholm, Paraxial light and atom optics: the optical Schrod̈inger equation and beyond. Phys. Rev. A 56, 2940 (1997)ADSCrossRefGoogle Scholar
  6. 6.
    S. Cruz y Cruz, O. Rosas-Ortiz, Leaky modes of waveguides as a classical optics analogy of quantum resonances. Adv. Math. Phys. 2015, 281472 (2015)Google Scholar
  7. 7.
    D. Gloge, D. Marcuse, Formal quantum theory of light rays. J. Opt. Soc. Am. 59, 1629 (1969)ADSCrossRefGoogle Scholar
  8. 8.
    G. Nienhuis, J. Visser, Angular momentum and vortices in paraxial beams. J. Opt. A: Pure Appl. Opt. 6, S248 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    S. Cruz y Cruz, Z. Gress, Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium. Ann. Phys. 383, 257 (2017)Google Scholar
  10. 10.
    A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986)Google Scholar
  11. 11.
    J. Guerrero, F.F. López-Ruiz, V. Aldaya, F. Cossío, Harmonic states for the free particle. J. Phys. A: Math. Theor. 44, 445307 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Guerrero, F. F. López-Ruiz, The quantum Arnold transformation and the Ermakov–Pinney equation. Phys. Scr. 87 038105 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    J. Guerrero, F.F. López-Ruiz, On the Lewis–Riesenfeld (Dodonov–Man’ko) invariant method. Phys. Scr. 90 074046 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    A. Perelomov, Generalized Coherent States and their Applications (Springer, Berlin, 1986)CrossRefGoogle Scholar
  15. 15.
    R.R. Puri, SU(m, n) coherent states in the bosonic representation and their generation in optical parametric processes. Phys. Rev. A 50, 5309 (1994)Google Scholar
  16. 16.
    P. Shanta, S. Chaturvedi, V. Srinivasan, G.S. Agarwal, C.L. Mehta, Unified approach to multiphoton coherent states. Phys. Rev. Lett. 72, 1447 (1994)ADSCrossRefGoogle Scholar
  17. 17.
    R.R. Puri, G.S. Agarwal, SU(1,  1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances. Phys. Rev.A 53, 1786 (1996)Google Scholar
  18. 18.
    I. Dhand, B.C. Sanders, H. de Guise, Algorithms for SU(n) boson realizations and \(\mathcal {D}\)-functions. J. Math. Phys. 56, 111705 (2015)Google Scholar
  19. 19.
    J. Schwinger, Quantum Theory of Angular Momentum (Academic, New York, 1965), pp. 229–279Google Scholar
  20. 20.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, Washington, DC, 1970)zbMATHGoogle Scholar
  21. 21.
    J. Negro, L.M. Nieto, O. Rosas-Ortiz, Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41,7964 (2000)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    O. Castaños, D. Schuch, O. Rosas-Ortiz, Generalized coherent states for time-dependent and nonlinear Hamiltonian operators via complex Riccati equations. J. Phys. A: Math. Theor. 46, 075304 ( 2013)ADSMathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universidad Autónoma del Estado de HidalgoCiudad del ConocimientoHidalgoMexico
  2. 2.Instituto Politécnico NacionalUPIITACiudad de MéxicoMexico

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