Abstract
We investigate the asymptotic behavior of large eigenvalue for the two-photon Rabi Hamiltonian, i.e., for the two-photon Jaynes–Cummings model without the rotating wave approximation. We prove that the spectrum of this Hamiltonian consists of two eigenvalue sequences \((E_n^+)_{n=0}^\infty \), \((E_n^-)_{n=0}^\infty \), satisfying the same two-term asymptotic formula with remainder \({{\,\mathrm{O}\,}}(n^{-1/3})\) when n tends to infinity. We also propose a conjecture on a three-term asymptotic formula modeled on the GRWA for the one-photon Rabi model.
Dedicated to the memory of Erik Balslev
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References
A. Boutet de Monvel, S. Naboko, L.O. Silva, The asymptotic behavior of eigenvalues of a modified Jaynes–Cummings model. Asymptot. Anal. 47(3–4), 291–315 (2006)
A. Boutet de Monvel, L. Zielinski, Asymptotic behavior of large eigenvalues of a modified Jaynes–Cummings model, in Spectral Theory and Differential Equations, American Mathematical Society Translations: Series 2, vol. 233 (Amer. Math. Soc., Providence, RI, 2014), pp. 77–93
A. Boutet de Monvel, L. Zielinski, Asymptotic behavior of large eigenvalues for Jaynes–Cummings type models. J. Spectr. Theory 7(2), 559–631 (2017)
A. Boutet de Monvel, L. Zielinski, Oscillatory behavior of large eigenvalues in quantum Rabi models. Int. Math. Res. Not. IMRN 59 pp. (2021) (to appear). https://arXiv.org/abs/1711.03366v2
A. Boutet de Monvel, L. Zielinski, On the spectrum of the quantum Rabi model, in Analysis as a Tool in Mathematical Physics: In Memory of Boris Pavlov. Operator Theory: Advances and Applications, vol. 276 (Birkhäuser/Springer, Cham), pp. 183–193
D. Braak, Q.-H. Chen, M.T. Batchelor, E. Solano, Semi-classical and quantum Rabi models: in celebration of 80 years [Preface]. J. Phys. A 49(30), 300301, 4 (2016)
P.A. Cojuhari, J. Janas, Discreteness of the spectrum for some unbounded Jacobi matrices. Acta Sci. Math. (Szeged) 73(3–4), 649–667 (2007)
L. Duan, Y.-F. Xie, D. Braak, Q.-H. Chen, Two-photon Rabi model: analytic solutions and spectral collapse. J. Phys. A 49(46), 464002, 13 (2016)
J. Edward, Spectra of Jacobi matrices, differential equations on the circle, and the \({\rm su}(1,1)\) Lie algebra. SIAM J. Math. Anal. 24(3), 824–831 (1993)
C. Emary, R.F. Bishop, Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians. J. Math. Phys. 43(8), 3916–3926 (2002)
C. Emary, R.F. Bishop, Exact isolated solutions for the two-photon Rabi Hamiltonian. J. Phys. A 35(39), 8231–8241 (2002)
I.D. Feranchuk, L.I. Komarov, A.P. Ulyanenkov, Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method. J. Phys. A: Math. Gen. 29(14), 4035–4047 (1996)
C.C. Gerry, Two-photon Jaynes–Cummings model interacting with the squeezed vacuum. Phys. Rev. A (3) 37(7), 2683–2686 (1988)
E.K. Irish, Generalized rotating-wave approximation for arbitrarily large coupling. Phys. Rev. Lett. 99(17), 173601 (2007)
J. Janas, M. Malejki, Alternative approaches to asymptotic behavior of eigenvalues of some unbounded Jacobi matrices. J. Comput. Appl. Math. 200(1), 342–356 (2007)
J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach. SIAM J. Math. Anal. 36(2), 643–658 (electronic) (2004)
E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51(1), 89–109 (1963)
M. Malejki, Asymptotics of the discrete spectrum for complex Jacobi matrices. Opuscula Math. 34(1), 139–160 (2014)
D.R. Masson, J. Repka, Spectral theory of Jacobi matrices in \(l^2({\bf Z})\) and the \({\rm su}(1,1)\) Lie algebra. SIAM J. Math. Anal. 22(4), 1131–1146 (1991)
I.I. Rabi, On the process of space quantization. Phys. Rev. 49(4), 324 (1936)
I.I. Rabi, Space quantization in a gyrating magnetic field. Phys. Rev. 51(8), 652 (1937)
Rozenbljum, G.V.: Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle (Russian). Trudy Maskov. Mat. Obshch 36, 59–84 (1978)
M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997)
È.A. Tur, Jaynes–Cummings model: solution without rotating wave approximation. Opt. Spectrosc. 89(4), 574–588 (2000)
E.A. Tur, Jaynes–Cummings model without rotating wave approximation. Asymptotics of eigenvalues, 12 pp. (2002). https://arXiv.org/abs/math-ph/0211055
Q. Xie, H. Zhong, M.T. Batchelor, C. Lee, The quantum Rabi model: solution and dynamics. J. Phys. A 50(11), 113001, 40 (2017)
E.A. Yanovich, Asymptotics of eigenvalues of an energy operator in a problem of quantum physics, in Operator Methods in Mathematical Physics. Operator Theory: Advances and Applications, vol. 227 (Birkhäuser/Springer Basel AG, Basel, 2013), pp. 165–177
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Boutet de Monvel, A., Zielinski, L. (2021). Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model. In: Albeverio, S., Balslev, A., Weder, R. (eds) Schrödinger Operators, Spectral Analysis and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-030-68490-7_5
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