Abstract

We examine the absorption of a weak probe beam in a laser-driven tripod-type atom with three closely lying ground levels, where both the driving and probe lasers interact simultaneously with the three transitions. The effects of spontaneously generated coherence (SGC) are taken into account. We introduce dipole moments in the dressed-state picture and the Hamiltonian in terms of the dressed states describing the interaction between the probe and the atom. Gain spectrum under various conditions are presented and analyzed. We show that the spectral structure and the gain amplitude of the probe are strongly influenced by the effect of SGC and the frequency separation of the three closely lying ground levels. The tripod-type atomic system with quantum interference in spontaneous emission can be simulated in the dressed-state picture of a coherently driven four-level N-type system where no SGC effect exists.

© 2012 Optical Society of America

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  1. G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
    [CrossRef]
  2. X. H. Yang and S. Y. Zhu, “Control of coherent population transfer via spontaneous decay-induced coherence,” Phys. Rev. A 77, 063822 (2008).
    [CrossRef]
  3. E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
    [CrossRef]
  4. E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
    [CrossRef]
  5. M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
    [CrossRef]
  6. A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
    [CrossRef]
  7. C. H. Raymond Ooi, “Effects of spontaneously generated coherence on two-photon correlation in a double-cascade scheme,” Phys. Rev. A 75, 043818 (2007).
    [CrossRef]
  8. J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
    [CrossRef]
  9. Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
    [CrossRef]
  10. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
    [CrossRef]
  11. G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000).
    [CrossRef]
  12. Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
    [CrossRef]
  13. J. H. Wu, A. J. Li, Yue Ding, Y. C. Zhao, and J. Y. Gao, “Control of spontaneous emission from a coherently driven four-level atom,” Phys. Rev. A 72, 023802 (2005).
    [CrossRef]
  14. A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
    [CrossRef]
  15. 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, 033816 (2006).
    [CrossRef]
  16. 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, 053806 (2008).
    [CrossRef]
  17. Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
    [CrossRef]
  18. C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
    [CrossRef]
  19. C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
    [CrossRef]
  20. S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
    [CrossRef]
  21. P. Zhou and S. Swain, “Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion,” Phys. Rev. Lett. 78, 832–835 (1997).
    [CrossRef]
  22. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
    [CrossRef]
  23. P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
    [CrossRef]
  24. S. Menon and G. S. Agarwal, “Effects of spontaneously generated coherence on the pump-probe response of a Λ system,” Phys. Rev. A 57, 4014–4018 (1998).
    [CrossRef]
  25. W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
    [CrossRef]
  26. J. H. Wu and J. Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
    [CrossRef]
  27. W. H. Xu and J. Y. Gao, “Gain spectrum of a laser-driven-type atom with vacuum-induced coherence,” J. Opt. Soc. Am. B 22, 2385–2392 (2005).
    [CrossRef]
  28. B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972).
    [CrossRef]
  29. 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, 1630–1649 (1990).
    [CrossRef]
  30. M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
    [CrossRef]
  31. H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
    [CrossRef]

2012 (1)

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

2011 (1)

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

2010 (1)

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

2009 (2)

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
[CrossRef]

2008 (5)

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

X. H. Yang and S. Y. Zhu, “Control of coherent population transfer via spontaneous decay-induced coherence,” Phys. Rev. A 77, 063822 (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, 053806 (2008).
[CrossRef]

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
[CrossRef]

2007 (1)

C. H. Raymond Ooi, “Effects of spontaneously generated coherence on two-photon correlation in a double-cascade scheme,” Phys. Rev. A 75, 043818 (2007).
[CrossRef]

2006 (2)

A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
[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, 033816 (2006).
[CrossRef]

2005 (4)

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

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[CrossRef]

M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

W. H. Xu and J. Y. Gao, “Gain spectrum of a laser-driven-type atom with vacuum-induced coherence,” J. Opt. Soc. Am. B 22, 2385–2392 (2005).
[CrossRef]

2004 (2)

H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

2002 (3)

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

J. H. Wu and J. Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

2001 (1)

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
[CrossRef]

2000 (2)

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
[CrossRef]

1999 (2)

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
[CrossRef]

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

1998 (1)

S. Menon and G. S. Agarwal, “Effects of spontaneously generated coherence on the pump-probe response of a Λ system,” Phys. Rev. A 57, 4014–4018 (1998).
[CrossRef]

1997 (1)

P. Zhou and S. Swain, “Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion,” Phys. Rev. Lett. 78, 832–835 (1997).
[CrossRef]

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, 1630–1649 (1990).
[CrossRef]

1972 (1)

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000).
[CrossRef]

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

S. Menon and G. S. Agarwal, “Effects of spontaneously generated coherence on the pump-probe response of a Λ system,” Phys. Rev. A 57, 4014–4018 (1998).
[CrossRef]

Antón, M. A.

M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

Bao, Q. Q.

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

Calderón, Oscar G.

M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

Carreño, F.

M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (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, 033816 (2006).
[CrossRef]

Chen, H.

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

Cui, C. L.

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

Cui, H. N.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

Ding, Yue

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

Dong, P.

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

Ficek, Z.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

Fountoulakis, A.

A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
[CrossRef]

Gao, J. W.

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

Gao, J. Y.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[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, 053806 (2008).
[CrossRef]

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
[CrossRef]

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[CrossRef]

W. H. Xu and J. Y. Gao, “Gain spectrum of a laser-driven-type atom with vacuum-induced coherence,” J. Opt. Soc. Am. B 22, 2385–2392 (2005).
[CrossRef]

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

J. H. Wu and J. Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

Hernandez, G.

H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

Jiang, Y.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
[CrossRef]

Kang, H.

H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

Kang, Z. H.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
[CrossRef]

Knight, P. L.

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
[CrossRef]

Kou, J.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

Kylstra, N. J.

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
[CrossRef]

Li, A. J.

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (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, 053806 (2008).
[CrossRef]

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

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[CrossRef]

Li, J. H.

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, 033816 (2006).
[CrossRef]

Li, Z. H.

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[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, 033816 (2006).
[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, 013807 (1999).
[CrossRef]

S. Menon and G. S. Agarwal, “Effects of spontaneously generated coherence on the pump-probe response of a Λ system,” Phys. Rev. A 57, 4014–4018 (1998).
[CrossRef]

Mollow, B. R.

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972).
[CrossRef]

Narducci, L. M.

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, 1630–1649 (1990).
[CrossRef]

Ooi, C. H. Raymond

C. H. Raymond Ooi, “Effects of spontaneously generated coherence on two-photon correlation in a double-cascade scheme,” Phys. Rev. A 75, 043818 (2007).
[CrossRef]

Oppo, G. L.

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, 1630–1649 (1990).
[CrossRef]

Paspalakis, E.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
[CrossRef]

A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
[CrossRef]

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, 033816 (2006).
[CrossRef]

Rickey, E. G.

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
[CrossRef]

Ru, P.

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, 1630–1649 (1990).
[CrossRef]

Scully, M. O.

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, 1630–1649 (1990).
[CrossRef]

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, 053806 (2008).
[CrossRef]

Swain, S.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

P. Zhou and S. Swain, “Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion,” Phys. Rev. Lett. 78, 832–835 (1997).
[CrossRef]

Tang, S. H.

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

Terzis, A. F.

A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
[CrossRef]

Tian, S. C.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

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, 1630–1649 (1990).
[CrossRef]

Vitanov, N. V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
[CrossRef]

Wan, R. G.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

Wang, C. L.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

C. L. Wang, A. J. Li, X. Y. Zhou, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Investigation of spontaneously generated coherence in dressed states of 85Rb atoms,” Opt. Lett. 33, 687–689 (2008).
[CrossRef]

Wang, D. W.

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

Wang, K.

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

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, 053806 (2008).
[CrossRef]

Wang, Lei

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[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, 053806 (2008).
[CrossRef]

Wu, J. H.

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[CrossRef]

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

J. H. Wu and J. Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

Xie, M.

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

Xu, J. P.

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

Xu, W. H.

W. H. Xu and J. Y. Gao, “Gain spectrum of a laser-driven-type atom with vacuum-induced coherence,” J. Opt. Soc. Am. B 22, 2385–2392 (2005).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

Yan, M.

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
[CrossRef]

Yang, G. J.

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

Yang, X. H.

X. H. Yang and S. Y. Zhu, “Control of coherent population transfer via spontaneous decay-induced coherence,” Phys. Rev. A 77, 063822 (2008).
[CrossRef]

Yang, Y. P.

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

Yannopapas, V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
[CrossRef]

Zhang, H.

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

Zhang, Z.

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

Zhao, Y. C.

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

Zheng, H.

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

Zhou, P.

P. Zhou and S. Swain, “Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion,” Phys. Rev. Lett. 78, 832–835 (1997).
[CrossRef]

Zhou, X. Y.

Zhu, S. Y.

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

X. H. Yang and S. Y. Zhu, “Control of coherent population transfer via spontaneous decay-induced coherence,” Phys. Rev. A 77, 063822 (2008).
[CrossRef]

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

Zhu, Y. F.

H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
[CrossRef]

Zubairy, M. S.

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

A. J. Li, J. Y. Gao, J. H. Wu, and Lei Wang, “Simulating spontaneously generated coherence in a four-level atomic system,” J. Phys. B 38, 3815–3823 (2005).
[CrossRef]

Opt. Commun. (1)

S. C. Tian, Z. H. Kang, C. L. Wang, R. G. Wan, J. Kou, H. Zhang, Y. Jiang, H. N. Cui, and J. Y. Gao, “Observation of spontaneously generated coherence on absorption in rubidium atomic beam,” Opt. Commun. 285, 294–299 (2012).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (22)

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

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

S. Menon and G. S. Agarwal, “Effects of spontaneously generated coherence on the pump-probe response of a Λ system,” Phys. Rev. A 57, 4014–4018 (1998).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

J. H. Wu and J. Y. Gao, “Phase control of light amplification without inversion in a Λ system with spontaneously generated coherence,” Phys. Rev. A 65, 063807 (2002).
[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972).
[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, 1630–1649 (1990).
[CrossRef]

M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A 64, 013412 (2001).
[CrossRef]

H. Kang, G. Hernandez, and Y. F. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[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, 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, 053806 (2008).
[CrossRef]

Z. H. Li, D. W. Wang, H. Zheng, S. Y. Zhu, and M. S. Zubairy, “Quantum interference due to energy shifts and its effect on spontaneous emission,” Phys. Rev. A 82, 050501(R)(2010).
[CrossRef]

C. L. Wang, Z. H. Kang, S. C. Tian, Y. Jiang, and J. Y. Gao, “Effect of spontaneously generated coherence on absorption in a V-type system: investigation in dressed states,” Phys. Rev. A 79, 043810 (2009).
[CrossRef]

G. J. Yang, M. Xie, Z. Zhang, and K. Wang, “Effect of vacuum-induced coherences on coherent population trapping of moving atoms,” Phys. Rev. A 77, 063825 (2008).
[CrossRef]

X. H. Yang and S. Y. Zhu, “Control of coherent population transfer via spontaneous decay-induced coherence,” Phys. Rev. A 77, 063822 (2008).
[CrossRef]

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

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

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802 (2000).
[CrossRef]

M. A. Antón, Oscar G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

A. Fountoulakis, A. F. Terzis, and E. Paspalakis, “Coherent phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73, 033811 (2006).
[CrossRef]

C. H. Raymond Ooi, “Effects of spontaneously generated coherence on two-photon correlation in a double-cascade scheme,” Phys. Rev. A 75, 043818 (2007).
[CrossRef]

J. W. Gao, Q. Q. Bao, R. G. Wan, C. L. Cui, and J. H. Wu, “Triple photonic band-gap structure dynamically induced in the presence of spontaneously generated coherence,” Phys. Rev. A 83, 053815 (2011).
[CrossRef]

Phys. Rev. Lett. (5)

Y. P. Yang, J. P. Xu, H. Chen, and S. Y. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100, 043601 (2008).
[CrossRef]

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103, 063602 (2009).
[CrossRef]

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84, 5500–5503 (2000).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82, 2079–2082 (1999).
[CrossRef]

P. Zhou and S. Swain, “Quantum interference in probe absorption: narrow resonances, transparency, and gain without population inversion,” Phys. Rev. Lett. 78, 832–835 (1997).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

(a) Tripod-type atomic system under consideration. (b) Arrangement of field polarization and dipole moments. θ is the angle between every two dipole moments. The polarization directions of both the driving and the probe fields are the same direction. And the angle between this orientation and each of the dipole moment is the same.

Fig. 2.
Fig. 2.

(a),(c) Allowed spontaneous transitions and (b),(d) detected transitions by the probe between dressed states of two neighboring manifolds in the case of the nondegenerate levels. For (a) and (b), p = 0 . For (c) and (d), p = 1 . The parameters are Δ c = 0 and Ω 2 = Ω 3 = Ω 4 .

Fig. 3.
Fig. 3.

Gain spectra of the weak probe for ω 23 = ω 34 = 0 . (a)  p = 0 , (b)  p = 1 . Other parameters are γ 23 = γ 24 = γ 34 = 0.01 γ , γ 12 = γ 13 = γ 14 = γ , Δ c = 0 , and Ω 2 = Ω 3 = Ω 4 = 10 γ .

Fig. 4.
Fig. 4.

Gain spectra of the weak probe for ω 23 = ω 23 = 5 γ , with different degrees of SGC: (a)  p = 0 , (b)  p = 0.8 , (c)  p = 0.917 , and (d)  p = 1 . Other parameters are the same as those in Fig. 3.

Fig. 5.
Fig. 5.

(a) Allowed spontaneous transitions and (b) detected transitions by the probe between dressed states of two neighboring manifolds. (The transitions for p = 0 and p = 1 are the same.) Other parameters are the same as those in Fig. 2.

Fig. 6.
Fig. 6.

Steady-state population of the dressed states as a function of p . ρ Ψ 1 Ψ 1 = ρ Ψ 4 Ψ 4 (blue dotted line); ρ Ψ 2 Ψ 2 = ρ Ψ 3 Ψ 3 (red solid line). Parameters are ω 23 = ω 34 = 5 γ . Other parameters are the same as those in Fig. 2.

Fig. 7.
Fig. 7.

Steady-state population of the dressed states as a function of p . ρ Ψ 1 Ψ 1 = ρ Ψ 4 Ψ 4 (blue dotted line); ρ Ψ 2 Ψ 2 = ρ Ψ 3 Ψ 3 (red solid line). Parameters are ω 23 = ω 34 = 20 γ . Other parameters are the same as those in Fig. 2.

Fig. 8.
Fig. 8.

Gain spectra of the weak probe for ω 23 = ω 34 = 20 γ , with different degrees of the SGC: (a)  p = 0 , (b)  p = 0.5 , (c)  p = 0.8 , and (d)  p = 1 . Other parameters are the same as those in Fig. 3.

Fig. 9.
Fig. 9.

(a),(b) Steady-state populations of the dressed states as a function of ω . ρ Ψ 1 Ψ 1 = ρ Ψ 4 Ψ 4 (blue dotted line); ρ Ψ 2 Ψ 2 = ρ Ψ 3 Ψ 3 (red solid line). For (a), p = 0 ; for (b), p = 1 . (c),(d) Probe spectra as functions of frequency separation ω and detuning frequency Δ p . For (c), p = 0 ; for (d), p = 1 . Other parameters are the same as those in Fig. 3.

Fig. 10.
Fig. 10.

(a) Four-level N-type system without SGC; (b) four-level tripod-type atomic system with SGC. The former one is equivalent to the latter one in the dressed-state representation of two additional fields ω 1 and ω 2 .

Equations (40)

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H c = - ω | 2 2 | + ω | 4 4 | + Ω ( | 1 2 | + | 1 3 | + | 1 4 | + H.c. ) .
ρ ˙ 22 = i Ω ( ρ 12 ρ 21 ) + γ ρ 11 + γ ( ρ 33 ρ 22 ) + γ ( ρ 44 ρ 22 ) ,
ρ ˙ 33 = i Ω ( ρ 13 ρ 31 ) + γ ρ 11 + γ ( ρ 22 ρ 33 ) + γ ( ρ 44 ρ 33 ) ,
ρ ˙ 44 = i Ω ( ρ 14 ρ 41 ) + γ ρ 11 + γ ( ρ 22 ρ 44 ) + γ ( ρ 33 ρ 44 ) ,
ρ ˙ 12 = i Ω ( ρ 11 ρ 22 ) + i Ω ( ρ 32 + ρ 42 ) ( 3 γ / 2 + i ω ) ρ 12 ,
ρ ˙ 13 = i Ω ( ρ 11 ρ 33 ) + i Ω ( ρ 23 + ρ 43 ) 3 γ ρ 13 / 2 ,
ρ ˙ 14 = i Ω ( ρ 11 ρ 44 ) + i Ω ( ρ 24 + ρ 34 ) ( 3 γ / 2 i ω ) ρ 14 ,
ρ ˙ 23 = i Ω ( ρ 13 ρ 21 ) ( i ω + γ ) ρ 23 + p γ ρ 11 ,
ρ ˙ 24 = i Ω ( ρ 14 ρ 21 ) ( 2 i ω + γ ) ρ 24 + p γ ρ 11 ,
ρ ˙ 34 = i Ω ( ρ 14 ρ 31 ) ( i ω + γ ) ρ 34 + p γ ρ 11 ,
ρ 11 + ρ 22 + ρ 33 + ρ 44 = 1 ,
ρ i j = ρ i j * .
A ( Δ p ) = Re 0 lim t [ P ( t + τ ) , P + ( t ) ] exp ( i Δ p τ ) d τ ,
d d t ψ = L ψ + I ,
ψ = ( ρ 12 , ρ 13 , ρ 14 , ρ 21 , ρ 22 , ρ 23 , ρ 24 , ρ 31 , ρ 32 , ρ 33 , ρ 34 , ρ 41 , ρ 42 , ρ 43 , ρ 44 , ) T .
I = ( i Ω , i Ω , i Ω , i Ω , γ , p γ , p γ , i Ω , p γ , γ , p γ , i Ω , p γ , p γ , γ ) T .
A ( Δ P ) = μ 2 Re { [ M 1 ( ρ 11 ρ 22 ) 1 + M 1 ρ 5 21 + M 1 ρ 9 31 + M 1 ρ 13 41 M 1 ρ 2 23 M 1 ρ 3 24 ] + p [ M 1 ( ρ 11 ρ 33 ) 2 + M 1 ρ 6 21 + M 1 ρ 10 31 + M 1 ρ 14 41 M 1 ρ 1 32 M 1 ρ 3 34 ] + p [ M 1 ( ρ 11 ρ 44 ) 3 + M 1 ρ 7 21 + M 1 ρ 11 31 + M 1 ρ 15 41 M 1 ρ 1 42 M 1 ρ 2 43 ] + [ M 2 ( ρ 11 ρ 33 ) 2 + M 2 ρ 6 21 + M 2 ρ 10 31 + M 2 ρ 14 41 M 2 ρ 1 32 M 2 ρ 3 34 ] + p [ M 2 ( ρ 11 ρ 22 ) 1 + M 2 ρ 5 21 + M 2 ρ 9 31 + M 2 ρ 13 41 M 2 ρ 2 23 M 2 ρ 3 24 ] + p [ M 2 ( ρ 11 ρ 44 ) 3 + M 2 ρ 7 21 + M 2 ρ 11 31 + M 2 ρ 15 41 M 2 ρ 1 42 M 2 ρ 2 43 ] + [ M 3 ( ρ 11 ρ 44 ) 3 + M 3 ρ 7 21 + M 3 ρ 11 31 + M 3 ρ 15 41 M 3 ρ 1 42 M 3 ρ 2 43 ] + p [ M 3 ( ρ 11 ρ 22 ) 1 + M 3 ρ 5 21 + M 3 ρ 9 31 + M 3 ρ 13 41 M 3 ρ 2 23 M 3 ρ 3 24 ] + p [ M 3 ( ρ 11 ρ 33 ) 2 + M 3 ρ 6 21 + M 2 ρ 10 31 + M 2 ρ 14 41 M 3 ρ 1 32 M 3 ρ 3 34 ] } ,
| Ψ i , N = C i 1 | 1 , N 1 + C i 2 | 2 , N + C i 3 | 3 , N + C i 4 | 4 , N ( i = 1 , 2 , 3 , 4 ) .
C i 1 = λ i ( λ i + ω ) ( λ i ω ) D i ,
C i 2 = λ i ( λ i ω ) Ω D i ,
C i 3 = ( λ i + ω ) ( λ i ω ) Ω D i ,
C i 4 = λ i ( λ i + ω ) Ω D i ,
D i = λ i 2 ( λ i ω ) 2 ( λ i + ω ) 2 + Ω 2 [ λ i 2 ( λ i ω ) 2 + ( λ i + ω ) 2 ( λ i ω ) 2 + λ i 2 ( λ i + ω ) 2 ] .
κ ± = 1 2 [ ( 3 Ω 2 + ω 2 ) ± ( 3 Ω 2 + ω 2 ) 2 4 ω 2 Ω 2 ] .
Ψ i , N + 1 | μ | Ψ j , N = C i 1 * C j 2 μ 12 + C i 1 * C j 3 μ 13 + C i 1 * C j 4 μ 14 .
H p = 1 2 μ E p [ e i ω p t ( | 1 2 | + | 1 3 | + | 1 4 | ) + H.c. ] = 1 2 μ E p ( e i ω p t i = 1 4 C i 1 | Ψ i j = 1 4 k = 2 4 C j k * Ψ j | + H.c. ) .
| Ψ 1 , N = 1 6 ( 3 | 1 , N 1 + | 2 , N + | 3 , N + | 4 , N ) ,
| Ψ 2 , N = 1 12 [ ( 3 1 ) | 2 , N + 2 | 3 , N + ( 3 1 ) | 4 , N ] ,
| Ψ 2 , N = 1 12 [ ( 3 1 ) | 2 , N + 2 | 3 , N + ( 3 1 ) | 4 , N ] ,
| Ψ 4 , N = 1 6 ( 3 | 1 , N 1 + | 2 , N + | 3 , N + | 4 , N ) .
Ψ 1 , N + 1 | μ | Ψ 1 , N = 3 6 ( μ 12 + μ 13 + μ 14 ) ,
Ψ 1 , N + 1 | μ | Ψ 2 , N = 6 12 [ ( 1 + 3 ) ( μ 13 μ 12 ) + ( 1 3 ) ( μ 13 μ 14 ) ] ,
Ψ 1 , N + 1 | μ | Ψ 3 , N = 6 12 [ ( 1 3 ) ( μ 13 μ 12 ) + ( 1 + 3 ) ( μ 13 μ 14 ) ] ,
Ψ 1 , N + 1 | μ | Ψ 4 , N = 3 6 ( μ 12 + μ 13 + μ 14 ) ,
Ψ 4 , N + 1 | μ | Ψ 1 , N = 3 6 ( μ 12 + μ 13 + μ 14 ) ,
Ψ 4 , N + 1 | μ | Ψ 2 , N = 6 12 [ ( 1 + 3 ) ( μ 13 μ 12 ) + ( 1 3 ) ( μ 13 μ 14 ) ] ,
Ψ 4 , N + 1 | μ | Ψ 3 , N = 6 12 [ ( 1 + 3 ) ( μ 13 μ 12 ) + ( 1 3 ) ( μ 13 μ 14 ) ] ,
Ψ 4 , N + 1 | μ | Ψ 4 , N = 3 6 ( μ 12 + μ 13 + μ 14 ) ,
Ψ i , N + 1 | μ | Ψ j , N = 0 ( i = 2 , 3 ; j = 1 , 2 , 3 , 4 ) .
H p = 1 2 μ E p [ e i ω p t ( | 1 2 | + | 1 3 | + | 1 4 | ) + H.c. ] = 3 3 μ E p [ e i ω p t ( | Ψ 1 | Ψ 4 ) ( Ψ 1 | + Ψ 4 | ) + H.c. ] .

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