Applications of Picard and Magnus expansions to the Rabi model

  • Fabrizio Angaroni
  • Giuliano Benenti
  • Giuliano Strini
Regular Article


We apply the Picard and Magnus expansions to both the semiclassical and the quantum Rabi model, with a switchable matter-field coupling. The case of the quantum Rabi model is a paradigmatic example of finite-time quantum electrodynamics (QED), and in this case we build an intuitive diagrammatic representation of the Picard series. In particular, we show that regular oscillations in the mean number of photons, ascribed to the dynamical Casimir effect (DCE) for the generation of photons and to the anti-DCE for their destruction, take place at twice the resonator frequency ω. Such oscillations, which are a clear dynamical “smoking gun” of the DCE and become clearly visible when the interaction strength enters ultrastrong coupling (USC) regime, can be predicted by first-order Picard expansion. We also show that the Magnus expansion can be used, through concatenation, as an efficient numerical integrator for both the semiclassical and the quantum Rabi model. In the first case, we find distinctive features in the Fourier spectrum of motion, with a single peak at the Rabi frequency Ω and doublets at frequencies 2 ± Ω, with n positive integer. We explain these doublets, which are a feature beyond the rotating wave approximation (RWA), on the basis of the Picard series.

Graphical abstract


Nonlinear Dynamics 


  1. 1.
    J. Bourassa, J.M. Gambetta, A.A. Abdumalikov, Jr., O. Astafiev, Y. Nakamura, A. Blais, Phys. Rev. A 80, 032109 (2009) CrossRefADSGoogle Scholar
  2. 2.
    T. Niemczyk, F. Deppe, H. Huebl, E. Menzel, F. Hocke, M.J. Schwarz, J.J. García-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, R. Gross, Nat. Phys. 6, 772 (2010) CrossRefGoogle Scholar
  3. 3.
    P. Forn-Díaz, J. Lisenfeld, D. Marcos, J.J. García-Ripoll, E. Solano, C.J.P.M. Harmans, J.E. Mooij, Phys. Rev. Lett. 105, 237001 (2010) CrossRefADSGoogle Scholar
  4. 4.
    P. Forn-Díaz, J.J. García-Ripoll, B. Peropadre, J.-L. Orgiazzi, M.A. Yurtalan, R. Belyansky, C.M. Wilson, A. Lupascu, Nat. Phys. 13, 39 (2017) CrossRefGoogle Scholar
  5. 5.
    F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, K. Semba, Nat. Phys. 13, 44 (2017) CrossRefGoogle Scholar
  6. 6.
    D. Markovič, S. Jezouin, Q. Ficheux, S. Fedortchenko, S. Felicetti, T. Coudreau, P. Milman, Z. Leghtas, B. Huard, Phys. Rev. Lett. 121, 040505 (2018) CrossRefADSGoogle Scholar
  7. 7.
    G.T. Moore, J. Math. Phys. (N.Y.) 11, 2679 (1970) CrossRefADSGoogle Scholar
  8. 8.
    S. De Liberato, D. Gerace, I. Carusotto, C. Ciuti, Phys. Rev. A 80, 053810 (2009) CrossRefADSGoogle Scholar
  9. 9.
    V.V. Dodonov, Phys. Scripta 82, 038105 (2010) CrossRefADSGoogle Scholar
  10. 10.
    P.D. Nation, J.R. Johansson, M.P. Blencowe, F. Nori, Rev. Mod. Phys. 84, 1 (2012) CrossRefADSGoogle Scholar
  11. 11.
    J.-C. Jaskula, G.B. Partridge, M. Bonneau, R. Lopes, J. Ruaudel, D. Boiron, C. I. Westbrook, Phys. Rev. Lett. 109, 220401 (2012) CrossRefADSGoogle Scholar
  12. 12.
    S. Koghee, M. Wouters, Phys. Rev. Lett. 112, 036406 (2014) CrossRefADSGoogle Scholar
  13. 13.
    S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, E. Solano, Phys. Rev. Lett. 113, 093602 (2014) CrossRefADSGoogle Scholar
  14. 14.
    C. Sabín, I. Fuentes, G. Johansson, Phys. Rev. A 92, 012314 (2015) CrossRefADSGoogle Scholar
  15. 15.
    C. Sabín, G. Adesso, Phys. Rev. A 92, 042107 (2015) CrossRefADSGoogle Scholar
  16. 16.
    R. Stassi, S. De Liberato, L. Garziano, B. Spagnolo, S. Savasta, Phys. Rev. A 92, 013830 (2015) CrossRefADSGoogle Scholar
  17. 17.
    G. Benenti, S. Siccardi, G. Strini, Eur. Phys. J. D 68, 139 (2014) CrossRefADSGoogle Scholar
  18. 18.
    G. Benenti, A. D’Arrigo, S. Siccardi, G. Strini, Phys. Rev. A 90, 052313 (2014) CrossRefADSGoogle Scholar
  19. 19.
    G. Benenti, G. Strini, Phys. Rev. A 91, 020502(R) (2015) MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    F. Hoeb, F. Angaroni, J. Zoller, T. Calarco, G. Strini, S. Montangero, G. Benenti, Phys. Rev. A 96, 033851 (2017) CrossRefADSGoogle Scholar
  21. 21.
    C.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori, P. Delsing, Nature (London) 479, 376 (2011) CrossRefADSGoogle Scholar
  22. 22.
    P. Lähteenmäki, G.S. Paraoanu, J. Hassel, P.J. Hakonen, PNAS 110, 4234 (2013) CrossRefADSGoogle Scholar
  23. 23.
    K. Nomoto, R. Fukuda, Progr. Theor. Phys. 86, 269 (1991) CrossRefADSGoogle Scholar
  24. 24.
    P. Meystre, M. Sargent III, Elements of quantum optics, 4th Ed. (Springer–Verlag, Berlin, 2007) Google Scholar
  25. 25.
    D. Braak, Q.-H. Chen, M.T. Batchelor, E. Solano, J. Phys. A 49, 300301 (2016) MathSciNetCrossRefGoogle Scholar
  26. 26.
    I.M. de Sousa, A.V. Dodonov, J. Phys. A: Math. Theor. 48, 245302 (2015) CrossRefADSGoogle Scholar
  27. 27.
    D.S. Veloso, A.V. Dodonov, J. Phys. B 48, 165503 (2015) CrossRefADSGoogle Scholar
  28. 28.
    A. Motazedifard, M.H. Naderi, R. Roknizadeh, J. Opt. Soc. Am. B 32, 1555 (2015) CrossRefADSGoogle Scholar
  29. 29.
    S. Blanes, F. Casas, J.A. Oteo, J. Ros, Phys. Rep. 470, 151 (2009) MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    P.C. Moan, J. Niesen, J. Found. Comp. Math. 8, 291 (2008) CrossRefGoogle Scholar
  31. 31.
    F. Casas, J. Phys. A 40, 15001 (2007) MathSciNetCrossRefADSGoogle Scholar
  32. 32.
    N.W. McLachlan, Theory and Applications of Mathieu Functions (Dover Publications, New York, 1964) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Fabrizio Angaroni
    • 1
    • 2
  • Giuliano Benenti
    • 1
    • 2
    • 3
  • Giuliano Strini
    • 4
  1. 1.Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’InsubriaComoItaly
  2. 2.Istituto Nazionale di Fisica Nucleare, Sezione di MilanoMilanoItaly
  3. 3.NEST, Istituto Nanoscienze-CNRPisaItaly
  4. 4.Department of PhysicsUniversity of MilanMilanoItaly

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