Applications of Picard and Magnus expansions to the Rabi model

  • Fabrizio Angaroni
  • Giuliano BenentiEmail author
  • 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) CrossRefGoogle 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) CrossRefGoogle 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) CrossRefGoogle Scholar
  7. 7.
    G.T. Moore, J. Math. Phys. (N.Y.) 11, 2679 (1970) CrossRefGoogle Scholar
  8. 8.
    S. De Liberato, D. Gerace, I. Carusotto, C. Ciuti, Phys. Rev. A 80, 053810 (2009) CrossRefGoogle Scholar
  9. 9.
    V.V. Dodonov, Phys. Scripta 82, 038105 (2010) CrossRefGoogle Scholar
  10. 10.
    P.D. Nation, J.R. Johansson, M.P. Blencowe, F. Nori, Rev. Mod. Phys. 84, 1 (2012) CrossRefGoogle 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) CrossRefGoogle Scholar
  12. 12.
    S. Koghee, M. Wouters, Phys. Rev. Lett. 112, 036406 (2014) CrossRefGoogle Scholar
  13. 13.
    S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, E. Solano, Phys. Rev. Lett. 113, 093602 (2014) CrossRefGoogle Scholar
  14. 14.
    C. Sabín, I. Fuentes, G. Johansson, Phys. Rev. A 92, 012314 (2015) CrossRefGoogle Scholar
  15. 15.
    C. Sabín, G. Adesso, Phys. Rev. A 92, 042107 (2015) CrossRefGoogle Scholar
  16. 16.
    R. Stassi, S. De Liberato, L. Garziano, B. Spagnolo, S. Savasta, Phys. Rev. A 92, 013830 (2015) CrossRefGoogle Scholar
  17. 17.
    G. Benenti, S. Siccardi, G. Strini, Eur. Phys. J. D 68, 139 (2014) CrossRefGoogle Scholar
  18. 18.
    G. Benenti, A. D’Arrigo, S. Siccardi, G. Strini, Phys. Rev. A 90, 052313 (2014) CrossRefGoogle Scholar
  19. 19.
    G. Benenti, G. Strini, Phys. Rev. A 91, 020502(R) (2015) MathSciNetCrossRefGoogle Scholar
  20. 20.
    F. Hoeb, F. Angaroni, J. Zoller, T. Calarco, G. Strini, S. Montangero, G. Benenti, Phys. Rev. A 96, 033851 (2017) CrossRefGoogle 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) CrossRefGoogle Scholar
  22. 22.
    P. Lähteenmäki, G.S. Paraoanu, J. Hassel, P.J. Hakonen, PNAS 110, 4234 (2013) CrossRefGoogle Scholar
  23. 23.
    K. Nomoto, R. Fukuda, Progr. Theor. Phys. 86, 269 (1991) CrossRefGoogle 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) CrossRefGoogle Scholar
  27. 27.
    D.S. Veloso, A.V. Dodonov, J. Phys. B 48, 165503 (2015) CrossRefGoogle Scholar
  28. 28.
    A. Motazedifard, M.H. Naderi, R. Roknizadeh, J. Opt. Soc. Am. B 32, 1555 (2015) CrossRefGoogle Scholar
  29. 29.
    S. Blanes, F. Casas, J.A. Oteo, J. Ros, Phys. Rep. 470, 151 (2009) MathSciNetCrossRefGoogle 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) MathSciNetCrossRefGoogle 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
    Email author
  • 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

Personalised recommendations