Opuscula Math. 41, no. 3 (2021), 301-333
https://doi.org/10.7494/OpMath.2021.41.3.301

Opuscula Mathematica

# Perturbation series for Jacobi matrices and the quantum Rabi model

Mirna Charif
Lech Zielinski

Abstract. We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.

Keywords: Jacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, quantum Rabi model, rotating wave approximation.

Mathematics Subject Classification: 81Q10, 47B36, 15A18.

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1. P.K. Aravind, J.O. Hirschfelder, Two-state systems in semiclassical and quantized fields, J. Phys. Chem. 88 (1984), no. 21, 4788-4801.
2. S.S. Bharadwaj, R.U. Haq, T.A. Wan, An explicit method for Schrieffer-Wolff transformation, arXiv:1901.08617.
3. A. Boutet de Monvel, L. Zielinski, On the spectrum of the quantum Rabi model, [in:] Analysis as a Tool in Mathematical Physics, Springer, 2020, 183-193.
4. D. Braak, Q.-H. Chen, M.T. Batchelor, E. Solano, Semi-classical and quantum Rabi models: in celebration of 80 years, J. of Physics A 49 (2016), 300301.
5. P.A. Cojuhari, J. Janas, Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged) 73 (2007), no. 3-4, 649-667.
6. M. Frasca, Third-order correction to localization in a two-level driven system, Phys. Rev. B 71 (2005), 073301.
7. S. He, Q.-H. Chen, X.-Z. Ren, T. Liu, K.-L. Wang, First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model, Phys. Rev. A 86 (2012), no. 3, 033837.
8. S. He, Y.-Y. Zhang, Q.-H. Chen, X.-Z. Ren, T. Liu, K.-L. Wang, Unified analytical treatments of qubit-oscillator systems, Chinese Physics B 22 (2013), no. 6, 064205.
9. J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2004), no. 2, 643-658.
10. E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51 (1963), no. 1, 89-109.
11. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1995.
12. D.J. Klein, Degenerate perturbation theory, J. Chem. Phys. 61 (1974), no. 3, 786.
13. P.O. Lödwing, A note on the quantum-mechanical perturbation theory, J. Chem. Phys. 19 (1951), no. 11, 1396.
14. I.I. Rabi, On the process of space quantization, Phys. Rev. 49 (1936), 324.
15. I.I. Rabi, Space quantization in a gyrating magnetic field, Phys. Rev. 51 (1937), 652.
16. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, 1978.
17. F. Rellich, J. Berkowitz, Perturbation Theory of Eigenvalue Problems, Gordon and Breach, 1969.
18. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Classics in Applied Mathematics, vol. 66, Manchester University Press, 1992.
19. M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge, 1997.
20. J.H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time, Phys. Rev. 138 (1965), B979.
21. È.A. Tur, Jaynes-Cummings model: solution without rotating wave approximation, Optics and Spectroscopy 89 (2000), no. 4, 574-588.
22. J.H. Van Vleck, On $$\sigma$$-type doubling and electron spin in the spectra of diatomic molecules, Phys. Rev. 33 (1929), 467.
23. Q. Xie, H. Zhong, T.M. Batchelor, C. Lee, The quantum Rabi model: solution and dynamics, J. Phys. A: Math. Theor. 50 (2017), 113001.
• Mirna Charif
• https://orcid.org/0000-0002-4403-1453
• Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville EA 2597, F-62228 Calais, France
• Lebanese University, Faculty of Sciences, Department of Mathematics, P.O. Box 826 Tripoli, Lebanon
• Lech Zielinski (corresponding author)
• https://orcid.org/0000-0002-7314-7586
• Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville EA 2597, F-62228 Calais, France
• Communicated by P.A. Cojuhari.
• Received: 2020-12-02.
• Revised: 2021-01-17.
• Accepted: 2021-01-23.
• Published online: 2021-04-19.

Cite this article as:
Mirna Charif, Lech Zielinski, Perturbation series for Jacobi matrices and the quantum Rabi model, Opuscula Math. 41, no. 3 (2021), 301-333, https://doi.org/10.7494/OpMath.2021.41.3.301

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