Abstract

One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.

© 2014 Optical Society of America

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  1. F. W. Cummings, “Stimulated emission of radiation in a single mode,” Phys. Rev. 140(4A), 1051–1056 (1965).
    [Crossref]
  2. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to beam maser,” Proc. IEEE 51(1), 89–109 (1963).
    [Crossref]
  3. M. Tavis and F. W. Cummings, “Approximate solutions for an n-molecule-radiation-field hamiltonian,” Phys. Rev. 188(2), 692–695 (1969).
    [Crossref]
  4. T. V. Foerster, “Comparison of quantum and semiclassical theories of interaction between a 2-level atom and radiation-field,” J. Phys. Math. Gen. 8(1), 95–102 (1975).
    [Crossref]
  5. Saifullah, “Feedback control of probability amplitudes for two-level atom in optical field,” Opt. Commun. 281(4), 640–643 (2008).
    [Crossref]
  6. S. V. Prants and V. Y. Sirotkin, “Effects of the Rabi oscillations on the atomic motion in a standing-wave cavity,” Phys. Rev. A 64(3), 033412 (2001).
    [Crossref]
  7. V. Y. Sirotkin and S. V. Prants, “Random walking of a two-level atom in a standing-wave field,” Proc. SPIE 4750, 97–103 (2002).
    [Crossref]
  8. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
    [Crossref]
  9. S. Borisenok and S. Khalid, “Linear Feed forward Control of Two-Level Quantum System by Modulated External Field,” Opt. Commun. 284(14), 3562–3567 (2011).
    [Crossref]
  10. S. Khalid, “Two-Level Decay Quantum System with Open-Loop Control Optical Field,” J. Opt. Soc. Am. B 30(2), 428–430 (2013).
    [Crossref]
  11. C. C. Chang and L. Lin, “Light-mediated quantum phase transition and manipulations of the quantum states of arrayed two-level atoms,” New J. Phys. 14(7), 073018 (2012).
    [Crossref]
  12. B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40(7), 1195–1238 (1993).
    [Crossref]
  13. M. D. Crisp, “Application of the displaced oscillator basis in quantum optics,” Phys. Rev. A 46(7), 4138–4149 (1992).
    [Crossref] [PubMed]
  14. J. Gea-Banacloche, “Jaynes-Cummings model with quasiclassical fields - the effect of dissipation,” Phys. Rev. A 47(3), 2221–2234 (1993).
    [Crossref] [PubMed]
  15. W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
    [Crossref]
  16. M. Jelenska-Kuklinska and M. Kus, “Resonance overlap in the semiclassical Jaynes-Cummings model,” Quantum Opt. 5(1), 25–31 (1993).
    [Crossref]
  17. B. W. Shore, “Coherent manipulations of atoms using laser light,” Acta Phys. Slovaca 58(3), 243–486 (2008).
    [Crossref]
  18. W. H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons, 1973).
  19. M. Sargent III, M. O. Scully and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).
  20. G. W. Ford and R. F. O. O’Connell, “The rotating wave approximation (RWA) of quantum optics: serious defect,” Physica A 243(3–4), 377–381 (1997).
    [Crossref]
  21. J. Seke, “Spontaneous emission of a 2-level system and the influence of the rotating-wave approximation on the final-state,” J. Stat. Phys. 33(1), 223–229 (1983).
    [Crossref]
  22. I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
    [Crossref]
  23. Y. Wu and X. X. Yang, “Strong-coupling theory of periodically driven two-level systems,” Phys. Rev. Lett. 98(1), 013601 (2007).
    [Crossref] [PubMed]
  24. I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51(8), 652–654 (1937).
    [Crossref]
  25. Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
    [Crossref]
  26. M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
    [Crossref]
  27. I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
    [Crossref] [PubMed]
  28. U. Herzog and J. A. Bergou, “Reflection of the Jaynes-Cummings dynamics in the spectrum of a regularly pumped micromaser,” Phys. Rev. A 55(2), 1385–1390 (1997).
    [Crossref]

2013 (1)

2012 (1)

C. C. Chang and L. Lin, “Light-mediated quantum phase transition and manipulations of the quantum states of arrayed two-level atoms,” New J. Phys. 14(7), 073018 (2012).
[Crossref]

2011 (1)

S. Borisenok and S. Khalid, “Linear Feed forward Control of Two-Level Quantum System by Modulated External Field,” Opt. Commun. 284(14), 3562–3567 (2011).
[Crossref]

2008 (2)

Saifullah, “Feedback control of probability amplitudes for two-level atom in optical field,” Opt. Commun. 281(4), 640–643 (2008).
[Crossref]

B. W. Shore, “Coherent manipulations of atoms using laser light,” Acta Phys. Slovaca 58(3), 243–486 (2008).
[Crossref]

2007 (1)

Y. Wu and X. X. Yang, “Strong-coupling theory of periodically driven two-level systems,” Phys. Rev. Lett. 98(1), 013601 (2007).
[Crossref] [PubMed]

2002 (1)

V. Y. Sirotkin and S. V. Prants, “Random walking of a two-level atom in a standing-wave field,” Proc. SPIE 4750, 97–103 (2002).
[Crossref]

2001 (1)

S. V. Prants and V. Y. Sirotkin, “Effects of the Rabi oscillations on the atomic motion in a standing-wave cavity,” Phys. Rev. A 64(3), 033412 (2001).
[Crossref]

1997 (3)

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

U. Herzog and J. A. Bergou, “Reflection of the Jaynes-Cummings dynamics in the spectrum of a regularly pumped micromaser,” Phys. Rev. A 55(2), 1385–1390 (1997).
[Crossref]

G. W. Ford and R. F. O. O’Connell, “The rotating wave approximation (RWA) of quantum optics: serious defect,” Physica A 243(3–4), 377–381 (1997).
[Crossref]

1993 (3)

B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40(7), 1195–1238 (1993).
[Crossref]

J. Gea-Banacloche, “Jaynes-Cummings model with quasiclassical fields - the effect of dissipation,” Phys. Rev. A 47(3), 2221–2234 (1993).
[Crossref] [PubMed]

M. Jelenska-Kuklinska and M. Kus, “Resonance overlap in the semiclassical Jaynes-Cummings model,” Quantum Opt. 5(1), 25–31 (1993).
[Crossref]

1992 (2)

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

M. D. Crisp, “Application of the displaced oscillator basis in quantum optics,” Phys. Rev. A 46(7), 4138–4149 (1992).
[Crossref] [PubMed]

1991 (1)

W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
[Crossref]

1987 (1)

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

1983 (2)

J. Seke, “Spontaneous emission of a 2-level system and the influence of the rotating-wave approximation on the final-state,” J. Stat. Phys. 33(1), 223–229 (1983).
[Crossref]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

1979 (1)

I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
[Crossref]

1975 (1)

T. V. Foerster, “Comparison of quantum and semiclassical theories of interaction between a 2-level atom and radiation-field,” J. Phys. Math. Gen. 8(1), 95–102 (1975).
[Crossref]

1969 (1)

M. Tavis and F. W. Cummings, “Approximate solutions for an n-molecule-radiation-field hamiltonian,” Phys. Rev. 188(2), 692–695 (1969).
[Crossref]

1965 (1)

F. W. Cummings, “Stimulated emission of radiation in a single mode,” Phys. Rev. 140(4A), 1051–1056 (1965).
[Crossref]

1963 (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to beam maser,” Proc. IEEE 51(1), 89–109 (1963).
[Crossref]

1937 (1)

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51(8), 652–654 (1937).
[Crossref]

Armitage, A.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Bergou, J. A.

U. Herzog and J. A. Bergou, “Reflection of the Jaynes-Cummings dynamics in the spectrum of a regularly pumped micromaser,” Phys. Rev. A 55(2), 1385–1390 (1997).
[Crossref]

Borisenok, S.

S. Borisenok and S. Khalid, “Linear Feed forward Control of Two-Level Quantum System by Modulated External Field,” Opt. Commun. 284(14), 3562–3567 (2011).
[Crossref]

Chakravarty, S.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Chang, C. C.

C. C. Chang and L. Lin, “Light-mediated quantum phase transition and manipulations of the quantum states of arrayed two-level atoms,” New J. Phys. 14(7), 073018 (2012).
[Crossref]

Cohen, E.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Crisp, M. D.

M. D. Crisp, “Application of the displaced oscillator basis in quantum optics,” Phys. Rev. A 46(7), 4138–4149 (1992).
[Crossref] [PubMed]

Cummings, F. W.

M. Tavis and F. W. Cummings, “Approximate solutions for an n-molecule-radiation-field hamiltonian,” Phys. Rev. 188(2), 692–695 (1969).
[Crossref]

F. W. Cummings, “Stimulated emission of radiation in a single mode,” Phys. Rev. 140(4A), 1051–1056 (1965).
[Crossref]

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to beam maser,” Proc. IEEE 51(1), 89–109 (1963).
[Crossref]

Dorsey, A. T.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Euler, N.

W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
[Crossref]

Fisher, M. P. A.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Fisher, T. A.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Foerster, T. V.

T. V. Foerster, “Comparison of quantum and semiclassical theories of interaction between a 2-level atom and radiation-field,” J. Phys. Math. Gen. 8(1), 95–102 (1975).
[Crossref]

Ford, G. W.

G. W. Ford and R. F. O. O’Connell, “The rotating wave approximation (RWA) of quantum optics: serious defect,” Physica A 243(3–4), 377–381 (1997).
[Crossref]

Garg, A.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Gea-Banacloche, J.

J. Gea-Banacloche, “Jaynes-Cummings model with quasiclassical fields - the effect of dissipation,” Phys. Rev. A 47(3), 2221–2234 (1993).
[Crossref] [PubMed]

Goy, P.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

Gross, M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

Haroche, S.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

Herzog, U.

U. Herzog and J. A. Bergou, “Reflection of the Jaynes-Cummings dynamics in the spectrum of a regularly pumped micromaser,” Phys. Rev. A 55(2), 1385–1390 (1997).
[Crossref]

Ikonen, E.

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to beam maser,” Proc. IEEE 51(1), 89–109 (1963).
[Crossref]

Jelenska-Kuklinska, M.

M. Jelenska-Kuklinska and M. Kus, “Resonance overlap in the semiclassical Jaynes-Cummings model,” Quantum Opt. 5(1), 25–31 (1993).
[Crossref]

Jortner, J.

I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
[Crossref]

Kaluzny, Y.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

Katila, T.

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

Khalid, S.

S. Khalid, “Two-Level Decay Quantum System with Open-Loop Control Optical Field,” J. Opt. Soc. Am. B 30(2), 428–430 (2013).
[Crossref]

S. Borisenok and S. Khalid, “Linear Feed forward Control of Two-Level Quantum System by Modulated External Field,” Opt. Commun. 284(14), 3562–3567 (2011).
[Crossref]

Knight, P. L.

B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40(7), 1195–1238 (1993).
[Crossref]

Kus, M.

M. Jelenska-Kuklinska and M. Kus, “Resonance overlap in the semiclassical Jaynes-Cummings model,” Quantum Opt. 5(1), 25–31 (1993).
[Crossref]

Leggett, A. J.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Lin, L.

C. C. Chang and L. Lin, “Light-mediated quantum phase transition and manipulations of the quantum states of arrayed two-level atoms,” New J. Phys. 14(7), 073018 (2012).
[Crossref]

Lindén, J.

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

Lippmaa, M.

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

Mulser, P.

W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
[Crossref]

O’Connell, R. F. O.

G. W. Ford and R. F. O. O’Connell, “The rotating wave approximation (RWA) of quantum optics: serious defect,” Physica A 243(3–4), 377–381 (1997).
[Crossref]

Opher-Lipson, M.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Prants, S. V.

V. Y. Sirotkin and S. V. Prants, “Random walking of a two-level atom in a standing-wave field,” Proc. SPIE 4750, 97–103 (2002).
[Crossref]

S. V. Prants and V. Y. Sirotkin, “Effects of the Rabi oscillations on the atomic motion in a standing-wave cavity,” Phys. Rev. A 64(3), 033412 (2001).
[Crossref]

Rabi, I. I.

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51(8), 652–654 (1937).
[Crossref]

Raimond, J. M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, and S. Haroche, “Observation of self-induced Rabi oscillations in 2-level atoms excited inside a resonant cavity - the ringing regime of super-radiance,” Phys. Rev. Lett. 51(13), 1175–1178 (1983).
[Crossref]

Roberts, J. S.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Sage, M. L.

I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
[Crossref]

Saifullah,

Saifullah, “Feedback control of probability amplitudes for two-level atom in optical field,” Opt. Commun. 281(4), 640–643 (2008).
[Crossref]

Schek, I.

I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
[Crossref]

Seke, J.

J. Seke, “Spontaneous emission of a 2-level system and the influence of the rotating-wave approximation on the final-state,” J. Stat. Phys. 33(1), 223–229 (1983).
[Crossref]

Shore, B. W.

B. W. Shore, “Coherent manipulations of atoms using laser light,” Acta Phys. Slovaca 58(3), 243–486 (2008).
[Crossref]

B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40(7), 1195–1238 (1993).
[Crossref]

Sirotkin, V. Y.

V. Y. Sirotkin and S. V. Prants, “Random walking of a two-level atom in a standing-wave field,” Proc. SPIE 4750, 97–103 (2002).
[Crossref]

S. V. Prants and V. Y. Sirotkin, “Effects of the Rabi oscillations on the atomic motion in a standing-wave cavity,” Phys. Rev. A 64(3), 033412 (2001).
[Crossref]

Skolnick, M. S.

M. Opher-Lipson, E. Cohen, A. Armitage, M. S. Skolnick, T. A. Fisher, and J. S. Roberts, “Magnetic field effect on the Rabi-split modes in GaAs microcavities,” Phys. Status Solidi A 164(1), 35–38 (1997).
[Crossref]

Steeb, W. H.

W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
[Crossref]

Tavis, M.

M. Tavis and F. W. Cummings, “Approximate solutions for an n-molecule-radiation-field hamiltonian,” Phys. Rev. 188(2), 692–695 (1969).
[Crossref]

Tittonen, I.

I. Tittonen, M. Lippmaa, E. Ikonen, J. Lindén, and T. Katila, “Observation of mössbauer resonance line splitting caused by Rabi oscillations,” Phys. Rev. Lett. 69(19), 2815–2818 (1992).
[Crossref] [PubMed]

Wu, Y.

Y. Wu and X. X. Yang, “Strong-coupling theory of periodically driven two-level systems,” Phys. Rev. Lett. 98(1), 013601 (2007).
[Crossref] [PubMed]

Yang, X. X.

Y. Wu and X. X. Yang, “Strong-coupling theory of periodically driven two-level systems,” Phys. Rev. Lett. 98(1), 013601 (2007).
[Crossref] [PubMed]

Zwerger, W.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative 2-state system,” Rev. Mod. Phys. 59(1), 1–85 (1987).
[Crossref]

Acta Phys. Slovaca (1)

B. W. Shore, “Coherent manipulations of atoms using laser light,” Acta Phys. Slovaca 58(3), 243–486 (2008).
[Crossref]

Chem. Phys. Lett. (1)

I. Schek, M. L. Sage, and J. Jortner, “Validity of the rotating wave approximation for high-order molecular multi-photon processes,” Chem. Phys. Lett. 63(2), 230–235 (1979).
[Crossref]

J. Math. Phys. (1)

W. H. Steeb, N. Euler, and P. Mulser, “Semiclassical Jaynes-Cummings model, painleve test, and exact-solutions,” J. Math. Phys. 32(12), 3405–3406 (1991).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic of the interaction of a single two-level atom with a single-mode optical field.

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Fig. 2
Fig. 2 The probability of the atom being in state | b 〉 oscillates with the frequency of Ω 1 .

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Fig. 3
Fig. 3 The comparison of the numerical solution (black curve), the rotating wave approximation (RWA) (blue curve) and the analytical solution (red curve) of the probability of the atom being in upper state | b 〉 at time t .

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Fig. 4
Fig. 4 Comparison of the analytical solution (red curve) against the numerical solution (black curve) for different detuned systems with varying Rabi frequency. ω b a = 1 in panels (a), (b) and (c), ω b a = 10 in panels (d), (e) and (f), ω b a = 20 in panels (g), (h) and (i), Ω a b = 1 in panels (a), (d) and (g), Ω a b = 0.5 in panels (b), (e) and (h), while Ω a b = 0.1 in panels (c), (f) and (i).

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Fig. 5
Fig. 5 The influence of the initial phase φ = 0 (black curve) and φ = 0.5 π (red curve) on the probability of finding the atom in the upper level state.

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Fig. 6
Fig. 6 Rabi splitting in the spectrum for the probability of finding the atom in the upper level state under the rotating wave approximation for a system with ω b a = 20 , ω = 10 , Ω a b = 1. (a) φ = 0 , (b) φ = 0.5 π .

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Fig. 7
Fig. 7 Rabi splitting in the spectrum for the probability of finding the atom in the upper level state without the rotating wave approximation for a system with ω b a = 20 , ω = 10 , Ω a b = 1. (a) φ = 0 , (b) φ = 0.5 π .

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Fig. 8
Fig. 8 (a) The influence of the initial phase φ = 0 (black curve) and φ = 0.5 π (red curve) on the probability of finding the atom in the upper level state. Rabi splitting in the spectrum for the probability of finding the atom in the upper level state without the rotating wave approximation for a system with ω b a = 13 , ω = 10 , Ω a b = 1. (b) φ = 0 , (c) φ = 0.5 π .

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Equations (31)

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E ( t ) = E 0 cos ( ω t + φ ) .
| ψ ( t ) 〉 = C a ( t ) | a 〉 + C b ( t ) | b 〉 ,
| ψ ˙ ( t ) 〉 = − i H | ψ ( t ) 〉 / ℏ
H = H 0 + H 1 ,
H 0 = ℏ ω a | a 〉 〈 a | + ℏ ω b | b 〉 〈 b | ,
H 1 = − ( D a b | a 〉 〈 b | + D b a | b 〉 〈 a | ) E ( t ) ,
c a . ( t ) = i Ω a b c b ( e − i ( ω b a − ω ) t + i φ + e − i ( ω b a + ω ) t − i φ ) /2 ,
c b . ( t ) = i Ω a b c a ( e i ( ω b a − ω ) t − i φ + e i ( ω b a + ω ) t + i φ ) /2 ,
Ω a b = | D b a | E 0 / ℏ ,
c a . ( t ) = i Ω a b c b e − i ( ω b a − ω ) t + i φ / 2 ,
c b . ( t ) = i Ω a b c a e i ( ω b a − ω ) t − i φ /2 .
Ω 1 = ( ω ba − ω ) 2 + Ω a b 2 ,
c b ( t )= i ( Ω a b / Ω 1 ) e i ( ω b a − ω ) t / 2 − i φ sin ( Ω 1 t / 2 ) .
| c b ( t ) | 2 = c b c b * =( Ω ab / Ω 1 ) 2 sin 2 ( Ω 1 t / 2 ) ,
| c a ( t ) | 2 = 1 − | c b ( t ) | 2 =1 − ( Ω ab / Ω 1 ) 2 sin 2 ( Ω 1 t / 2 ) ,
| c a ( t ) | 2 = cos 2 ( Ω 1 t / 2 ) ,
| c b ( t ) | 2 = sin 2 ( Ω 1 t / 2 ) .
c a . ( t ) = i Ω a b c b e − i ( ω b a + ω ) t − i φ / 2 ,
c b . ( t ) = i Ω a b c a e i ( ω b a + ω ) t + i φ / 2 ,
c b ( t ) = i ( Ω ab / Ω 2 )e i ( ω b a + ω ) t / 2 + i φ sin ( Ω 2 t / 2 ) ,
| c b ( t ) | 2 =( Ω ab / Ω 2 ) 2 sin 2 ( Ω 2 t / 2 ) ,
| c a ( t ) | 2 = 1 − | c b ( t ) | 2 =1 − ( Ω ab / Ω 2 ) 2 sin 2 ( Ω 2 t / 2 ) ,
Ω 2 = ( ω b a + ω ) 2 + Ω a b 2 .
c b ( t ) =c b 1 ( t ) +c b 2 ( t ) = i ( Ω ab Ω 1 ) e i ( ω b a − ω ) t / 2 − i φ sin ( Ω 1 t 2 ) + i ( Ω ab Ω 2 ) e i ( ω b a + ω ) t / 2 + i φ sin ( Ω 2 t 2 ) .              
c b ( t ) =c b 1 ( t ) +c b 2 ( t ) = i ( Ω ab / Ω 1 ) e i ( ω b a − ω ) t / 2 − i φ sin ( Ω 1 t / 2 ) + i ( Ω ab / Ω 2 ) e i ( ω b a + ω ) t / 2 + i φ sin ( Ω 2 t / 2 ) ,
| c b ( t ) | 2 = ( Ω ab Ω 1 ) 2 sin 2 ( Ω 1 t 2 ) + ( Ω ab Ω 2 ) 2 sin 2 ( Ω 2 t 2 ) +2 Ω ab Ω 1 Ω ab Ω 2 sin ( Ω 1 t 2 ) sin ( Ω 2 t 2 ) cos ( ω t +2 φ ) .
c b ( t ) = i ( Ω ab / Ω 1 ) e i ( ω b a − ω ) t / 2 − i φ sin ( Ω 1 t / 2 ) ,
c b ( ω ) = i Ω a b 2 Ω 1 e − i φ { δ [ ω − ( ω ba − ω 2 + Ω 1 2 ) ] − δ [ ω − ( ω ba − ω 2 − Ω 1 2 ) ] } .
c b ( t ) =c b 1 ( t ) +c b 2 ( t ) = i ( Ω ab / Ω 1 ) e i ( ω b a − ω ) t / 2 − i φ sin ( Ω 1 t / 2 ) + i ( Ω ab / Ω 2 ) e i ( ω b a + ω ) t / 2 + i φ sin ( Ω 2 t / 2 ) ,
c b ( ω ) = i Ω a b 2 Ω 1 e − i φ { δ [ ω − ( ω ba − ω 2 + Ω 1 2 ) ] − δ [ ω − ( ω ba − ω 2 − Ω 1 2 ) ] } + i Ω a b 2 Ω 2 e i φ { δ [ ω − ( ω ba + ω 2 + Ω 2 2 ) ] − δ [ ω − ( ω ba + ω 2 − Ω 2 2 ) ] } ,
ω 1 = ω ba − ω 2 + Ω 1 2 , ω 2 = ω ba − ω 2 − Ω 1 2 , ω 3 = ω ba + ω 2 − Ω 2 2 , ω 4 = ω ba + ω 2 + Ω 2 2 .

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