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Analytical Solutions of Basic Models in Quantum Optics

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Applications + Practical Conceptualization + Mathematics = fruitful Innovation

Part of the book series: Mathematics for Industry ((MFI,volume 11))

Abstract

The recent progress in the analytical solution of models invented to describe theoretically the interaction of matter with light on an atomic scale is reviewed. The methods employ the classical theory of linear differential equations in the complex domain (Fuchsian equations). The linking concept is provided by the Bargmann Hilbert space of analytic functions, which is isomorphic to \(L^2(\mathbb {R})\), the standard Hilbert space for a single continuous degree of freedom in quantum mechanics. I give the solution of the quantum Rabi model in some detail and sketch the solution of its generalization, the asymmetric Dicke model. Characteristic properties of the respective spectra are derived directly from the singularity structure of the corresponding system of differential equations.

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References

  1. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atom-Photon Interactions: Basic Processes and Applications. Wiley, Weinheim (2004)

    Google Scholar 

  2. Haroche, S., Raymond, J.M.: Exploring the Quantum. Oxford University Press, New York (2006)

    Book  MATH  Google Scholar 

  3. Romero, G., Ballester, D., Wang, Y.M., Scarani, V., Solano, E.: Ultrafast quantum gates in circuit QED. Phys. Rev. Lett. 108, 120501 (2012)

    Article  Google Scholar 

  4. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  5. Wallraff, A., Schuster, D.I., Blais, A., Frunzio, L., Huang, R.S., Majer, J., Kumar, S., Girvin, S.M., Schoelkopf, R.J.: Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162 (2004)

    Article  Google Scholar 

  6. Niemczyk, T., Deppe, F., Huebl, H., Menzel, E.P., Hocke, F.: Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 6, 772 (2010)

    Article  Google Scholar 

  7. Forn-Diaz, P., Lisenfeld, J., Marcos, D., García-Ripoll, J.J., Solano, E., Harmans, C.J.P.M., Mooij, J.E.: Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime. Phys. Rev. Lett. 105, 237001 (2010)

    Article  Google Scholar 

  8. Rabi, I.I.: On the process of space quantization. Phys. Rev. 49, 324 (1936)

    Article  MATH  Google Scholar 

  9. Jaynes, E.T., Cummings, F.W.: Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89 (1963)

    Article  Google Scholar 

  10. Klimov, A.B., Sainz, I., Chumakov, S.M.: Resonance expansion versus rotating-wave approximation. Phys. Rev. A 68, 063811 (2003)

    Article  Google Scholar 

  11. Amico, L., Frahm, H., Osterloh, A., Ribeiro, G.A.P.: Integrable spin-boson models descending from rational six-vertex models. Nucl. Phys. B 787, 283 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)

    Book  Google Scholar 

  13. Caux, J.S., Mossel, J.: Remarks on the notion of quantum integrability. J. Stat. Mech. P02023 (2011)

    Google Scholar 

  14. Casanova, J., Romero, G., Lizuain, I., García-Ripoll, J.J., Solano, E.: Deep strong coupling regime of the Jaynes-Cummings model. Phys. Rev. Lett. 105, 263603 (2010)

    Article  Google Scholar 

  15. Braak, D.: Integrability of the Rabi model. Phys. Rev. Lett. 107, 100401 (2011)

    Article  Google Scholar 

  16. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, part I. Commun. Pure Appl. Math. 14, 187 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  17. Braak, D.: Continued fractions and the Rabi model. J. Phys. A: Math. Theor. 46, 175301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)

    MATH  Google Scholar 

  19. Slavyanov, S.Y., Lay, W.: Special Functions. A Unified Theory Based on Singularities. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  20. Braak, D.: A generalized G-function for the quantum Rabi model. Ann. Phys. (Berlin) 525, L23 (2013)

    Google Scholar 

  21. Zhong, H., Xie, Q., Batchelor, M.T., Lee, C.: Analytical eigenstates for the quantum Rabi model. J. Phys. A: Math. Theor. 46, 14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ronveaux, A. (ed.): Heun’s Differential Equations. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  23. Maciejewski, A.J., Przybylska, M., Stachowiak, T.: Full spectrum of the Rabi model. Phys. Lett. A 378, 16 (2014)

    Article  MathSciNet  Google Scholar 

  24. Turbiner, A.V.: Quasi-exactly-solvable problems and \(sl(2)\) algebra. Commun. Math. Phys. 118, 467 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wakayama, M., Yamasaki, T.: The quantum Rabi model and the Lie algebra representations of \(\mathfrak{sl}_2\). J. Phys. A: Math. Theor. 47, 335203 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schweber, S.: On the application of Bargmann Hilbert spaces to dynamical problems. Ann. Phys., NY 41, 205 (1967)

    Google Scholar 

  27. Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9, 24 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  28. Swain, S.: A continued fraction solution to the problem of a single atom interacting with a single radiation mode in the electric dipole approximation. J. Phys. A: Math. Nucl. Gen. 6, 192 (1973)

    Article  Google Scholar 

  29. Tur, E.A.: Energy spectrum of the Hamiltonian of the Jaynes-Cummings model without rotating-wave approximation. Opt. Spectrosc. 91, 899 (2001)

    Article  Google Scholar 

  30. Dicke, R.H.: Coherence in spontaneous radiation processes. Phys. Rev. 93, 99 (1954)

    Article  MATH  Google Scholar 

  31. Tavis, M., Cummings, F.W.: Exact solution for an N-molecule-radiation-field Hamiltonian. Phys. Rev. 170, 379 (1968)

    Article  Google Scholar 

  32. Haack, G., Helmer, F., Mariantoni, M., Marquardt, F., Solano, E.: Resonant quantum gates in circuit quantum electrodynamics. Phys. Rev. B 82, 024514 (2010)

    Article  Google Scholar 

  33. Peng, J., Ren, Z.Z., Braak, D., Guo, G.J., Ju, G.X., Zhang, X., Guo, X.Y.: Solution of the two-qubit quantum Rabi model and its exceptional eigenstates. J. Phys. A: Math. Theor. 47, 265303 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Chilingaryan, S.A., Rodríguez-Lara, B.M.: The quantum Rabi model for two qubits. J. Phys. A: Math. Theor. 46, 335301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Braak, D.: Solution of the Dicke model for \(N=3\). J. Phys. B: At. Mol. Opt. Phys. 46, 224007 (2013)

    Article  MathSciNet  Google Scholar 

  36. Batchelor, M.T., Zhou, H.Q.: Integrability versus exact solvability in the quantum Rabi and Dicke models. Phys. Rev. A 91, 053808 (2015)

    Article  Google Scholar 

  37. Travěnec, I.: Solvability of the two-photon Rabi Hamiltonian. Phys. Rev. A 85, 043805 (2012)

    Article  Google Scholar 

  38. Fan, J., Yang, Z., Zhang, Y., Ma, J., Chen, G., Jia, S.: Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model. Phys. Rev. A 89, 023812 (2014)

    Article  Google Scholar 

  39. Wakayama, M.: Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun differential equations, eigenstates degeneration and the Rabi model. Kyushu University, MI-preprint series (2013)

    MATH  Google Scholar 

  40. Ichinose, T., Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Commun. Math. Phys. 258, 697 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hirokawa, M.: The Dicke-type crossing among eigenvalues of differential operators in a class of non-commutative oscillators. Indiana Univ. Math. J. 58, 1493 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wakayama, M.: Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation. Proc. Jpn. Acad. Ser. A 89, 69 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hiroshima, F., Sasaki, I.: Spectral analysis of non-commutative harmonic oscillators: the lowest eigenvalue and no crossing. J. Math. Anal. Appl. 105, 595 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hirokawa, M., Hiroshima, F.: Absence of energy level crossing for the ground state energy of the Rabi model. Comm. Stoch. Anal. (to appear)

    Google Scholar 

  45. Xie, Q.T., Cui, S., Cao, J.P., Amico, A., Fan, H.: Anisotropic Rabi model. Phys. Rev. X 4, 021046 (2014)

    Google Scholar 

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Acknowledgments

This work was supported by Deutsche Forschungsgemeinschaft through TRR 80.

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Authors and Affiliations

  1. EP VI and Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, Augsburg, Germany

    Daniel Braak

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  1. Daniel Braak
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Correspondence to Daniel Braak .

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Editors and Affiliations

  1. CSIRO Computational Informatics, Canberra, Aust Capital Terr, Australia

    Robert S. Anderssen

  2. La Trobe University, Melbourne, Victoria, Australia

    Philip Broadbridge

  3. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Yasuhide Fukumoto

  4. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Kenji Kajiwara

  5. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

  6. Leiden University, Leiden, The Netherlands

    Evgeny Verbitskiy

  7. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Masato Wakayama

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Braak, D. (2016). Analytical Solutions of Basic Models in Quantum Optics. In: Anderssen, R., et al. Applications + Practical Conceptualization + Mathematics = fruitful Innovation. Mathematics for Industry, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55342-7_7

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  • DOI: https://doi.org/10.1007/978-4-431-55342-7_7

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-55341-0

  • Online ISBN: 978-4-431-55342-7

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