Abstract

We demonstrate coherent control of spontaneous emission from an indirectly coupled transition in a microwave driven four-level atomic system. The transition of concern is not directly coupled by any laser fields, while the ground state is coupled to another ground state by a microwave field. We show that the coupling of the microwave field produces interesting features such as double narrow lines in the emission spectrum. The heights, widths and positions of the emission peaks can be controlled by modifying the Rabi frequency and detuning of the microwave field. We discuss the spectra in the dressed states basis.

© 2012 OSA

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    [CrossRef] [PubMed]
  3. V. V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17(7), 5774–5782 (2009).
    [CrossRef] [PubMed]
  4. D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
    [CrossRef] [PubMed]
  5. S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989).
    [CrossRef] [PubMed]
  6. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
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  7. S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
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  8. G. A. Koganov, B. Shif, and R. Shuker, “Field-driven super/subradiant lasing without inversion in three-level ladder scheme,” Opt. Lett. 36(15), 2779–2781 (2011).
    [CrossRef] [PubMed]
  9. A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
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  10. O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
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  11. M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
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  12. R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011).
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  26. E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998).
    [CrossRef]
  27. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996).
    [CrossRef] [PubMed]
  28. I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
    [CrossRef]
  29. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008).
    [CrossRef]
  30. L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
    [CrossRef] [PubMed]
  31. A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
    [CrossRef] [PubMed]
  32. D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
    [CrossRef] [PubMed]
  33. J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
    [CrossRef]
  34. F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
    [CrossRef]
  35. J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
    [CrossRef]
  36. A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
    [CrossRef]
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    [CrossRef]
  39. S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981).
    [CrossRef]
  40. C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977).
    [CrossRef]

2011 (7)

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
[CrossRef] [PubMed]

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011).
[CrossRef]

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011).
[CrossRef]

G. A. Koganov, B. Shif, and R. Shuker, “Field-driven super/subradiant lasing without inversion in three-level ladder scheme,” Opt. Lett. 36(15), 2779–2781 (2011).
[CrossRef] [PubMed]

X. D. Zeng, M. Z. Yu, D. W. Wang, J. P. Xu, and Y. P. Yang, “Spontaneous emission spectrum of a V-type three-level atom in a Fabry-Perot cavity containing left-handed materials,” J. Opt. Soc. Am. B 28(9), 2253–2259 (2011).
[CrossRef]

2010 (2)

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010).
[CrossRef] [PubMed]

2009 (2)

W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009).
[CrossRef]

V. V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17(7), 5774–5782 (2009).
[CrossRef] [PubMed]

2008 (4)

S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008).
[CrossRef]

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
[CrossRef] [PubMed]

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008).
[CrossRef]

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

2007 (1)

2006 (3)

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006).
[CrossRef] [PubMed]

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006).
[CrossRef]

2005 (2)

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

2002 (1)

J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett. 89(16), 163601 (2002).
[CrossRef] [PubMed]

2001 (2)

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001).
[CrossRef] [PubMed]

E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett. 87(25), 253001 (2001).
[CrossRef] [PubMed]

2000 (3)

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
[CrossRef] [PubMed]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
[CrossRef]

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
[CrossRef]

1999 (1)

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[CrossRef]

1998 (1)

E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998).
[CrossRef]

1996 (3)

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

1991 (2)

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
[CrossRef] [PubMed]

1990 (1)

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

1989 (1)

S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989).
[CrossRef] [PubMed]

1981 (1)

S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981).
[CrossRef]

1977 (1)

C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977).
[CrossRef]

1968 (1)

M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev. 172(2), 350–361 (1968).
[CrossRef]

Agarwal, G. S.

S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008).
[CrossRef]

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001).
[CrossRef] [PubMed]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[CrossRef]

Antón, M.

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

Arun, R.

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008).
[CrossRef]

Barberis-Blostein, P.

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

Bennett, C. H.

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
[CrossRef] [PubMed]

Calderón, O.

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

Carmichael, H. J.

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

Carreño, F.

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

Chen, A. X.

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

Clemens, J. P.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977).
[CrossRef]

Das, S.

S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008).
[CrossRef]

Ding, C. L.

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

Ding, Y.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

DiVincenzo, D. P.

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
[CrossRef] [PubMed]

Doss, H. M.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

Evangelou, S.

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011).
[CrossRef]

Evers, J.

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006).
[CrossRef] [PubMed]

J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett. 89(16), 163601 (2002).
[CrossRef] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
[CrossRef]

Freimund, D. L.

Frishman, E.

E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett. 87(25), 253001 (2001).
[CrossRef] [PubMed]

Gao, J. Y.

R. G. Wan, J. Kou, L. Jiang, Y. Jiang, and J. Y. Gao, “Two-dimensional atom localization via controlled spontaneous emission from a driven tripod system,” J. Opt. Soc. Am. B 28(1), 10–17 (2011).
[CrossRef]

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

Gauthier, D. J.

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
[CrossRef] [PubMed]

Ghafoor, F.

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
[CrossRef]

Gonzalo, I.

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

Grünwald, P.

P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010).
[CrossRef] [PubMed]

Harman, Z.

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989).
[CrossRef] [PubMed]

Hollberg, L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Jiang, L.

Jiang, X. Q.

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

Jiang, Y.

Kalachev, A.

A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006).
[CrossRef]

Kapale, K. T.

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
[CrossRef] [PubMed]

Keitel, C. H.

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
[CrossRef] [PubMed]

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006).
[CrossRef] [PubMed]

J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett. 89(16), 163601 (2002).
[CrossRef] [PubMed]

Kiffner, M.

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006).
[CrossRef] [PubMed]

Kilin, S. Y.

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
[CrossRef] [PubMed]

Knell, R. O.

Knight, P. L.

E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998).
[CrossRef]

Koganov, G. A.

Kou, J.

Kröll, S.

A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006).
[CrossRef]

Lax, M.

M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev. 172(2), 350–361 (1968).
[CrossRef]

Li, A. J.

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

Li, J. H.

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

Liu, J. B.

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

Lu, Z. W.

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Manka, A. S.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

Matsko, A. B.

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
[CrossRef]

Menon, S.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[CrossRef]

Mossberg, T. W.

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
[CrossRef] [PubMed]

Narducci, L. M.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

Nikonov, D. E.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Norris, D. G.

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

Oppo, G. L.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

Orozco, L. A.

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

M. L. Terraciano, R. O. Knell, D. L. Freimund, L. A. Orozco, J. P. Clemens, and P. R. Rice, “Enhanced spontaneous emission into the mode of a cavity QED system,” Opt. Lett. 32(8), 982–984 (2007).
[CrossRef] [PubMed]

Paspalakis, E.

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011).
[CrossRef]

E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998).
[CrossRef]

Postavaru, O.

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
[CrossRef] [PubMed]

Qi, C. C.

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

Reynaud, S.

C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977).
[CrossRef]

Rice, P. R.

Robinson, H. G.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Ru, P.

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

Scully, M. O.

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
[CrossRef] [PubMed]

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001).
[CrossRef] [PubMed]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
[CrossRef]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

Shapiro, M.

E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett. 87(25), 253001 (2001).
[CrossRef] [PubMed]

Shif, B.

Shuker, R.

Song, X. L.

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

Sun, X. D.

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

Swain, S.

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996).
[CrossRef] [PubMed]

S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981).
[CrossRef]

Temnov, V. V.

Terraciano, M. L.

Tredicce, J. R.

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

Velichansky, V. L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Vogel, W.

P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010).
[CrossRef] [PubMed]

Walther, H.

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001).
[CrossRef] [PubMed]

Wan, R. G.

Wang, D. W.

Wang, L.

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

Wei, X. G.

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

Woggon, U.

Wu, J. H.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

Xu, J. P.

Yang, X. X.

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

Yang, Y. P.

Yannopapas, V.

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011).
[CrossRef]

Yu, M. Z.

Zeng, X. D.

Zhan, Z. M.

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

Zhang, B.

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

Zhang, W. X.

W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009).
[CrossRef]

Zhao, Y. C.

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

Zhou, P.

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996).
[CrossRef] [PubMed]

Zhu, S. Y.

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
[CrossRef]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996).
[CrossRef] [PubMed]

Zhu, Y. F.

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
[CrossRef] [PubMed]

Zhuang, J.

W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009).
[CrossRef]

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

Zubairy, M. S.

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

S. Swain, “Master equation derivation of quantum regression theorem,” J. Phys. A 14(10), 2577–2580 (1981).
[CrossRef]

J. Phys. B (1)

C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10(3), 345–363 (1977).
[CrossRef]

Nature (1)

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

M. Lax, “Quantum noise. XI. multitime correspondence between quantum and classical Stochastic Processes,” Phys. Rev. 172(2), 350–361 (1968).
[CrossRef]

Phys. Rev. A (16)

S. Das and G. S. Agarwal, “Photon-photon correlations as a probe of vacuum-induced coherence effects,” Phys. Rev. A 77(3), 033850 (2008).
[CrossRef]

I. Gonzalo, M. Antón, F. Carreño, and O. Calderón, “Squeezing in a Λ-type three-level atom via spontaneously generated coherence,” Phys. Rev. A 72(3), 033809 (2005).
[CrossRef]

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77(3), 033820 (2008).
[CrossRef]

L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce, “Spontaneous emission and absorption properties of a driven three-level system,” Phys. Rev. A 42(3), 1630–1649 (1990).
[CrossRef] [PubMed]

A. S. Manka, H. M. Doss, L. M. Narducci, P. Ru, and G. L. Oppo, “Spontaneous emission and absorption properties of a driven three-level system. II. The Λ and cascade models,” Phys. Rev. A 43(7), 3748–3763 (1991).
[CrossRef] [PubMed]

J. H. Wu, A. J. Li, Y. Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72(2), 023802 (2005).
[CrossRef]

F. Ghafoor, S. Y. Zhu, and M. S. Zubairy, “Amplitude and phase control of spontaneous emission,” Phys. Rev. A 62(1), 013811 (2000).
[CrossRef]

J. H. Li, J. B. Liu, A. X. Chen, and C. C. Qi, “Spontaneous emission spectra and simulating multiple spontaneous generation coherence in a five-level atomic medium,” Phys. Rev. A 74(3), 033816 (2006).
[CrossRef]

A. J. Li, X. L. Song, X. G. Wei, L. Wang, and J. Y. Gao, “Effects of spontaneously generated coherence in a microwave-driven four-level atomic system,” Phys. Rev. A 77(5), 053806 (2008).
[CrossRef]

A. Kalachev and S. Kröll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006).
[CrossRef]

W. X. Zhang and J. Zhuang, “Dynamical control of two-level system decay and long time freezing,” Phys. Rev. A 79(1), 012310 (2009).
[CrossRef]

X. Q. Jiang, B. Zhang, Z. W. Lu, and X. D. Sun, “Control of spontaneous emission from a microwave-field-coupled three-level Λ-type atom in photonic crystals,” Phys. Rev. A 83(5), 053823 (2011).
[CrossRef]

S. Evangelou, V. Yannopapas, and E. Paspalakis, “Simulating quantum interference in spontaneous decay near plasmonic nanostructures: Population dynamics,” Phys. Rev. A 83(5), 055805 (2011).
[CrossRef]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[CrossRef]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, “Quantum limit of optical magnetometry in the presence of ac Stark shifts,” Phys. Rev. A 62(1), 013808 (2000).
[CrossRef]

C. L. Ding, J. H. Li, Z. M. Zhan, and X. X. Yang, “Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system,” Phys. Rev. A 83(6), 063834 (2011).
[CrossRef]

Phys. Rev. Lett. (14)

D. G. Norris, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Observation of ground-state quantum beats in atomic spontaneous emission,” Phys. Rev. Lett. 105(12), 123602 (2010).
[CrossRef] [PubMed]

S. E. Harris, “Lasers without inversion: Interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62(9), 1033–1036 (1989).
[CrossRef] [PubMed]

S. Y. Kilin, K. T. Kapale, and M. O. Scully, “Lasing without inversion: counterintuitive population dynamics in the transient regime,” Phys. Rev. Lett. 100(17), 173601 (2008).
[CrossRef] [PubMed]

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, and V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76(21), 3935–3938 (1996).
[CrossRef] [PubMed]

O. Postavaru, Z. Harman, and C. H. Keitel, “High-precision metrology of highly charged ions via relativistic resonance fluorescence,” Phys. Rev. Lett. 106(3), 033001 (2011).
[CrossRef] [PubMed]

E. Paspalakis and P. L. Knight, “Phase control of spontaneous emission,” Phys. Rev. Lett. 81(2), 293–296 (1998).
[CrossRef]

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77(19), 3995–3998 (1996).
[CrossRef] [PubMed]

P. Grünwald and W. Vogel, “Entanglement in atomic resonance fluorescence,” Phys. Rev. Lett. 104(23), 233602 (2010).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76(3), 388–391 (1996).
[CrossRef] [PubMed]

G. S. Agarwal, M. O. Scully, and H. Walther, “Inhibition of decoherence due to decay in a continuum,” Phys. Rev. Lett. 86(19), 4271–4274 (2001).
[CrossRef] [PubMed]

E. Frishman and M. Shapiro, “Complete suppression of spontaneous decay of a manifold of states by infrequent interruptions,” Phys. Rev. Lett. 87(25), 253001 (2001).
[CrossRef] [PubMed]

J. Evers and C. H. Keitel, “Spontaneous-emission suppression on arbitrary atomic transitions,” Phys. Rev. Lett. 89(16), 163601 (2002).
[CrossRef] [PubMed]

D. J. Gauthier, Y. F. Zhu, and T. W. Mossberg, “Observation of linewidth narrowing due to coherent stabilization of quantum fluctuations,” Phys. Rev. Lett. 66(19), 2460–2463 (1991).
[CrossRef] [PubMed]

M. Kiffner, J. Evers, and C. H. Keitel, “Quantum interference enforced by time-energy complementarity,” Phys. Rev. Lett. 96(10), 100403 (2006).
[CrossRef] [PubMed]

Other (1)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), Chap. 10.

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Figures (4)

Fig. 1
Fig. 1

The energy scheme under consideration. (a) In the bare-state basis. (b) In the dressed-state basis of the two laser fields and the microwave field. (c) In the dressed- state basis of the laser field ω 1 .

Fig. 2
Fig. 2

Numerical results of the emission spectrum with different Rabi frequency of the microwave field. The parameters used are (a) Ω 3 =0, (b) Ω 3 =0.05γ, (c) Ω 3 =0.1γ, (d) Ω 3 =0.2γ, (e) Ω 3 =1.5γ, (f) Ω 3 =2γ.

Fig. 3
Fig. 3

Properties as a function of the Rabi frequency Ω 3 , (a) The eigen energies given by Eq. (2), (b) Steady-state population, (c) decay rates of |b and |c.

Fig. 4
Fig. 4

Numerical results of the emission spectrum with different detuning of the microwave field. The parameters used are (a) Δ=0.1 , (b) Δ=0.2γ , (c) Δ=0.4γ .

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

H I =Δ|44|[ Ω 1 |13|+ Ω 2 |23|+ Ω 3 |24|+H.c.].
ρ t = 1 i [H,ρ]+Lρ,
ρ ˙ 22 =i Ω 2 ( ρ 32 ρ 23 )+i Ω 3 ( ρ 42 ρ 24 )+ γ 32 ρ 33 + γ 12 ρ 11 + γ 24 ρ 44 ( γ 12 + γ 24 ) ρ 22 ρ ˙ 33 =i Ω 1 ( ρ 13 ρ 31 )+i Ω 2 ( ρ 23 ρ 32 )( γ 31 + γ 32 + γ 34 ) ρ 33 ρ ˙ 44 =i Ω 3 ( ρ 24 ρ 42 )+ γ 34 ρ 33 + γ 14 ρ 11 + γ 24 ρ 22 ( γ 12 + γ 14 ) ρ 44 ρ ˙ 12 =[ γ 12 ( γ 14 + γ 24 )/2] ρ 12 +i Ω 1 ρ 32 i Ω 2 ρ 13 i Ω 3 ρ 14 ρ ˙ 13 =[( γ 12 + γ 14 + γ 31 + γ 32 + γ 34 )/2] ρ 13 +i Ω 1 ( ρ 33 ρ 11 )i Ω 2 ρ 12 ρ ˙ 14 =[ γ 14 ( γ 12 + γ 24 )+iΔ] ρ 14 i Ω 1 ρ 34 i Ω 3 ρ 12 ρ ˙ 23 =[( γ 12 + γ 24 + γ 31 + γ 32 + γ 34 )/2] ρ 23 +i Ω 2 ( ρ 33 ρ 22 )+i Ω 3 ρ 43 ρ ˙ 24 =[ γ 24 ( γ 12 + γ 14 )/2+iΔ] ρ 24 +i Ω 3 ( ρ 44 ρ 22 )+i Ω 2 ρ 34 ρ ˙ 34 =[( γ 31 + γ 32 + γ 34 + γ 14 + γ 24 )/2+iΔ] ρ 34 +i Ω 1 ρ 14 +i Ω 2 ρ 24 i Ω 3 ρ 32 .
S(ω)=Re 0 lim t <Δ D + (τ+t) Δ D (t)> e iωt dτ.
D + (t)= μ 34 a 4 a 3 , D (t)= [ D + (t)] ,
d dt Ψ=LΨ+I
S( δ k )=Re{ M 14,12 ρ ¯ 31 + M 14,13 ρ ¯ 32 + M 14,14 ρ ¯ 33 + M 14,15 ρ ¯ 34 + l N 14,l I l ρ ¯ 34 },
M i,j = [ (zL) 1 | z=i δ k ] i,j , N i,j = [ L 1 (zL) 1 | z=i δ k ] i,j .
λ a = ( Ω 1 2 + Ω 2 2 + Ω 3 2 )+ ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) 2 4 Ω 1 2 Ω 3 2 2 λ b = ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) 2 4 Ω 1 2 Ω 3 2 2 λ c = ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) 2 4 Ω 1 2 Ω 3 2 2 λ d = ( Ω 1 2 + Ω 2 2 + Ω 3 2 )+ ( Ω 1 2 + Ω 2 2 + Ω 3 2 ) 2 4 Ω 1 2 Ω 3 2 2 .
|a= x 1 |1+ x 2 |2+ x 3 |3+ x 4 |4 |b= x 1 |1 x 2 |2+ x 3 |3+ x 4 |4 |c= x 1 |1 x 2 |2+ x 3 |3+ x 4 |4 |d= x 1 |1+ x 2 |2+ x 3 |3+ x 4 |4
x 1 = Ω 1 ( λ i 2 Ω 3 2 ) Ω 2 Ω 3 λ i 1+ ( λ i Ω 3 ) 2 + ( λ i 2 Ω 3 2 Ω 2 Ω 3 ) 2 + ( Ω 1 ( λ i 2 Ω 3 2 ) Ω 2 Ω 3 λ i ) 2 x 2 = λ i Ω 3 1+ ( λ i Ω 3 ) 2 + ( λ i 2 Ω 3 2 Ω 2 Ω 3 ) 2 + ( Ω 1 ( λ i 2 Ω 3 2 ) Ω 2 Ω 3 λ i ) 2 x 3 = λ i 2 Ω 3 2 Ω 2 Ω 3 1+ ( λ i Ω 3 ) 2 + ( λ i 2 Ω 3 2 Ω 2 Ω 3 ) 2 + ( Ω 1 ( λ i 2 Ω 3 2 ) Ω 2 Ω 3 λ i ) 2 x 4 = 1 1+ ( λ i Ω 3 ) 2 + ( λ i 2 Ω 3 2 Ω 2 Ω 3 ) 2 + ( Ω 1 ( λ i 2 Ω 3 2 ) Ω 3 Ω 2 λ i ) 2 ,
|i= C ik |k , (i=a, b, c, d; k=1, 2, 3, 4),
R ij = | j|P|i | 2 = | C j4 | 2 μ 43 2 | C i3 | 2 , (i, j=a, b, c, d),

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