Abstract

In this article, we have shown that atomic states can be engineered by tuning the coupling Rabi frequency for a system with $\mathcal {N}$-type configuration. Electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA) and Autler-Townes (AT) splitting have been observed experimentally in a four level $\mathcal{N}$-type atomic vapor of $^{85}Rb$ atoms in the hyperfine levels of $D_2$ transition. It has been shown that the response of the atomic medium can be tuned from highly transparent to highly absorptive in our case. The evolution of the atomic states from the dark state $|{D}\rangle$ to the non-coupled state $|{NC}\rangle$ has been studied with the partial dressed state approach, which makes the backbone of the modification of the atomic response. In addition, the transient solutions in the time domain and the steady state solution in the frequency domain have been studied. The population dynamics and the coherence contribution in each case have been analyzed by time dependent solutions. The experimentally observed steady line-shape profiles have been supported by the steady state solution of optical-Bloch equations considering the Maxwell-Boltzmann velocity distributions of the atoms. It has been observed that the crossover between the EIT and the AT splitting has been replaced by the interference contribution of the EIA in this $\mathcal {N}$-type system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
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2018 (2)

D. V. Brazhnikov, S. M. Ignatovich, V. I. Vishnyakov, M. N. Skvortsov, C. Andreeva, V. M. Entin, and I. I. Ryabtsev, “High-quality electromagnetically-induced absorption resonances in a buffer-gas-filled vapour cell,” Laser Phys. Lett. 15(2), 025701 (2018).
[Crossref]

M. Bhattarai, V. Bharti, and V. Natarajan, “Tuning of the hanle effect from eit to eia using spatially separated probe and control beams,” Sci. Rep. 8(1), 7525 (2018).
[Crossref]

2017 (3)

2016 (1)

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

2015 (2)

D. J. Whiting, E. Bimbard, J. Keaveney, M. A. Zentile, C. S. Adams, and I. G. Hughes, “Electromagnetically induced absorption in a nondegenerate three-level ladder system,” Opt. Lett. 40(18), 4289–4292 (2015).
[Crossref]

D. Bhattacharyya, A. Ghosh, A. Bandyopadhyay, S. Saha, and S. De, “Observation of electromagnetically induced transparency in six-level rb atoms and theoretical simulation of the observed spectra,” J. Phys. B: At., Mol. Opt. Phys. 48(17), 175503 (2015).
[Crossref]

2013 (1)

C. Zhu, C. Tan, and G. Huang, “Crossover from electromagnetically induced transparency to autler-townes splitting in open v-type molecular systems,” Phys. Rev. A 87(4), 043813 (2013).
[Crossref]

2011 (2)

J. Sheng, X. Yang, U. Khadka, and M. Xiao, “All-optical switching in an n-type four-level atom-cavity system,” Opt. Express 19(18), 17059–17064 (2011).
[Crossref]

P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning autler-townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107(16), 163604 (2011).
[Crossref]

2010 (2)

2009 (2)

M. G. Bason, A. K. Mohapatra, K. J. Weatherill, and C. S. Adams, “Narrow absorptive resonances in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 42(7), 075503 (2009).
[Crossref]

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009).
[Crossref]

2007 (2)

L. Kong, X. Tu, J. Wang, Y. Zhu, and M. Zhan, “Sub-doppler spectral resolution in a resonantly driven four-level coherent medium,” Opt. Commun. 269(2), 362–369 (2007).
[Crossref]

V. Shah, S. Knappe, P. D. Schwindt, and J. Kitching, “Subpicotesla atomic magnetometry with a microfabricated vapour cell,” Nat. Photonics 1(11), 649–652 (2007).
[Crossref]

2005 (2)

2004 (2)

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Sub-doppler and subnatural narrowing of an absorption line induced by interacting dark resonances in a tripod system,” Phys. Rev. A 69(6), 063802 (2004).
[Crossref]

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Atomic four-level $n$n systems,” Phys. Rev. A 69(5), 053818 (2004).
[Crossref]

2003 (2)

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003).
[Crossref]

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of coherence and to transfer of population,” Phys. Rev. A 67(3), 033807 (2003).
[Crossref]

2001 (2)

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
[Crossref]

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Electromagnetically induced absorption spectra depending on intensities and detunings of the coupling field in cs vapour,” J. Phys. B: At., Mol. Opt. Phys. 34(23), 4801–4808 (2001).
[Crossref]

2000 (1)

B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84(18), 4080–4083 (2000).
[Crossref]

1999 (4)

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59(6), 4732–4735 (1999).
[Crossref]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

T. Nakajima, “Population transfer in n-level systems assisted by dressing fields,” Phys. Rev. A 59(1), 559–568 (1999).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

1997 (1)

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[Crossref]

1996 (1)

O. Schmidt, R. Wynands, Z. Hussein, and D. Meschede, “Steep dispersion and group velocity below $\frac {c}{3000}$c3000 in coherent population trapping,” Phys. Rev. A 53(1), R27–R30 (1996).
[Crossref]

1995 (2)

A. S. Zibrov, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. G. Robinson, “Experimental demonstration of laser oscillation without population inversion via quantum interference in rb,” Phys. Rev. Lett. 75(8), 1499–1502 (1995).
[Crossref]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: Propagation dynamics,” Phys. Rev. Lett. 74(13), 2447–2450 (1995).
[Crossref]

1955 (1)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703–722 (1955).
[Crossref]

Adams, C. S.

D. J. Whiting, E. Bimbard, J. Keaveney, M. A. Zentile, C. S. Adams, and I. G. Hughes, “Electromagnetically induced absorption in a nondegenerate three-level ladder system,” Opt. Lett. 40(18), 4289–4292 (2015).
[Crossref]

M. G. Bason, A. K. Mohapatra, K. J. Weatherill, and C. S. Adams, “Narrow absorptive resonances in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 42(7), 075503 (2009).
[Crossref]

Akulshin, A. M.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59(6), 4732–4735 (1999).
[Crossref]

Alotaibi, H. M.

D. Wang, C. Liu, C. Xiao, J. Zhang, H. M. Alotaibi, B. C. Sanders, L.-G. Wang, and S. Zhu, “Strong coherent light amplification with double electromagnetically induced transparency coherences,” Sci. Rep. 7(1), 5796 (2017).
[Crossref]

An, K.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Electromagnetically induced absorption spectra depending on intensities and detunings of the coupling field in cs vapour,” J. Phys. B: At., Mol. Opt. Phys. 34(23), 4801–4808 (2001).
[Crossref]

Andreeva, C.

D. V. Brazhnikov, S. M. Ignatovich, V. I. Vishnyakov, M. N. Skvortsov, C. Andreeva, V. M. Entin, and I. I. Ryabtsev, “High-quality electromagnetically-induced absorption resonances in a buffer-gas-filled vapour cell,” Laser Phys. Lett. 15(2), 025701 (2018).
[Crossref]

Anisimov, P. M.

P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning autler-townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107(16), 163604 (2011).
[Crossref]

Autler, S. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703–722 (1955).
[Crossref]

Bae, I.-H.

Bandyopadhyay, A.

K. Islam, A. Bandyopadhyay, B. C. Das, S. Saha, S. De, and D. Bhattacharyya, “Splitting of electromagnetically induced absorption signal in a five-level v-type atomic system,” J. Opt. Soc. Am. B 34(12), 2550–2557 (2017).
[Crossref]

D. Bhattacharyya, A. Ghosh, A. Bandyopadhyay, S. Saha, and S. De, “Observation of electromagnetically induced transparency in six-level rb atoms and theoretical simulation of the observed spectra,” J. Phys. B: At., Mol. Opt. Phys. 48(17), 175503 (2015).
[Crossref]

Barreiro, S.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59(6), 4732–4735 (1999).
[Crossref]

Bason, M. G.

M. G. Bason, A. K. Mohapatra, K. J. Weatherill, and C. S. Adams, “Narrow absorptive resonances in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 42(7), 075503 (2009).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

Bharti, V.

M. Bhattarai, V. Bharti, and V. Natarajan, “Tuning of the hanle effect from eit to eia using spatially separated probe and control beams,” Sci. Rep. 8(1), 7525 (2018).
[Crossref]

Bhattacharyya, D.

K. Islam, A. Bandyopadhyay, B. C. Das, S. Saha, S. De, and D. Bhattacharyya, “Splitting of electromagnetically induced absorption signal in a five-level v-type atomic system,” J. Opt. Soc. Am. B 34(12), 2550–2557 (2017).
[Crossref]

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

D. Bhattacharyya, A. Ghosh, A. Bandyopadhyay, S. Saha, and S. De, “Observation of electromagnetically induced transparency in six-level rb atoms and theoretical simulation of the observed spectra,” J. Phys. B: At., Mol. Opt. Phys. 48(17), 175503 (2015).
[Crossref]

Bhattarai, M.

M. Bhattarai, V. Bharti, and V. Natarajan, “Tuning of the hanle effect from eit to eia using spatially separated probe and control beams,” Sci. Rep. 8(1), 7525 (2018).
[Crossref]

Bimbard, E.

Brazhnikov, D. V.

D. V. Brazhnikov, S. M. Ignatovich, V. I. Vishnyakov, M. N. Skvortsov, C. Andreeva, V. M. Entin, and I. I. Ryabtsev, “High-quality electromagnetically-induced absorption resonances in a buffer-gas-filled vapour cell,” Laser Phys. Lett. 15(2), 025701 (2018).
[Crossref]

Cartaleva, S.

Chakrabarti, S.

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

Cho, Y.-W.

Danzmann, K.

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
[Crossref]

Das, A.

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

Das, B. C.

K. Islam, A. Bandyopadhyay, B. C. Das, S. Saha, S. De, and D. Bhattacharyya, “Splitting of electromagnetically induced absorption signal in a five-level v-type atomic system,” J. Opt. Soc. Am. B 34(12), 2550–2557 (2017).
[Crossref]

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

De, S.

K. Islam, A. Bandyopadhyay, B. C. Das, S. Saha, S. De, and D. Bhattacharyya, “Splitting of electromagnetically induced absorption signal in a five-level v-type atomic system,” J. Opt. Soc. Am. B 34(12), 2550–2557 (2017).
[Crossref]

B. C. Das, D. Bhattacharyya, A. Das, S. Chakrabarti, and S. De, “Simultaneous observations of electromagnetically induced transparency (eit) and absorption (eia) in a multi-level v-type system of 87rb and theoretical simulation of the observed spectra using a multi-mode approach,” J. Chem. Phys. 145(22), 224312 (2016).
[Crossref]

D. Bhattacharyya, A. Ghosh, A. Bandyopadhyay, S. Saha, and S. De, “Observation of electromagnetically induced transparency in six-level rb atoms and theoretical simulation of the observed spectra,” J. Phys. B: At., Mol. Opt. Phys. 48(17), 175503 (2015).
[Crossref]

Dowling, J. P.

P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning autler-townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107(16), 163604 (2011).
[Crossref]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

Entin, V. M.

D. V. Brazhnikov, S. M. Ignatovich, V. I. Vishnyakov, M. N. Skvortsov, C. Andreeva, V. M. Entin, and I. I. Ryabtsev, “High-quality electromagnetically-induced absorption resonances in a buffer-gas-filled vapour cell,” Laser Phys. Lett. 15(2), 025701 (2018).
[Crossref]

Fioretti, A.

Fleischhauer, A.

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D. Wang, C. Liu, C. Xiao, J. Zhang, H. M. Alotaibi, B. C. Sanders, L.-G. Wang, and S. Zhu, “Strong coherent light amplification with double electromagnetically induced transparency coherences,” Sci. Rep. 7(1), 5796 (2017).
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D. Wang, C. Liu, C. Xiao, J. Zhang, H. M. Alotaibi, B. C. Sanders, L.-G. Wang, and S. Zhu, “Strong coherent light amplification with double electromagnetically induced transparency coherences,” Sci. Rep. 7(1), 5796 (2017).
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J. Chem. Phys. (1)

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

J. Phys. B: At., Mol. Opt. Phys. (3)

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Electromagnetically induced absorption spectra depending on intensities and detunings of the coupling field in cs vapour,” J. Phys. B: At., Mol. Opt. Phys. 34(23), 4801–4808 (2001).
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D. Bhattacharyya, A. Ghosh, A. Bandyopadhyay, S. Saha, and S. De, “Observation of electromagnetically induced transparency in six-level rb atoms and theoretical simulation of the observed spectra,” J. Phys. B: At., Mol. Opt. Phys. 48(17), 175503 (2015).
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M. G. Bason, A. K. Mohapatra, K. J. Weatherill, and C. S. Adams, “Narrow absorptive resonances in a four-level atomic system,” J. Phys. B: At., Mol. Opt. Phys. 42(7), 075503 (2009).
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Laser Phys. Lett. (1)

D. V. Brazhnikov, S. M. Ignatovich, V. I. Vishnyakov, M. N. Skvortsov, C. Andreeva, V. M. Entin, and I. I. Ryabtsev, “High-quality electromagnetically-induced absorption resonances in a buffer-gas-filled vapour cell,” Laser Phys. Lett. 15(2), 025701 (2018).
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Nat. Photonics (1)

V. Shah, S. Knappe, P. D. Schwindt, and J. Kitching, “Subpicotesla atomic magnetometry with a microfabricated vapour cell,” Nat. Photonics 1(11), 649–652 (2007).
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Opt. Commun. (2)

A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müller, and R.-H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997).
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M. Bhattarai, V. Bharti, and V. Natarajan, “Tuning of the hanle effect from eit to eia using spatially separated probe and control beams,” Sci. Rep. 8(1), 7525 (2018).
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Figures (8)

Fig. 1.
Fig. 1. Experimental setup to study the N-type system. ECDL: External cavity diode laser, BS: beam splitter, VND: variable neutral density filter, NPBS: Non polarizing cubic beam splitter, PBS: polarizing cubic beam splitter, AOM: acousto optic modulator, Rb Cell: Rubidium vapour cell.
Fig. 2.
Fig. 2. Experimentally observed probe transmission as a function of pump detunning. The pump Rabi frequency $\Omega _{c}$ was fixed at $17.57$ $MHz$ and the coupling Rabi frequency $\Omega _{b}$ was varied continuously (see legends).
Fig. 3.
Fig. 3. Width (EIT, EIA) and peak separations (AT) vs coupling Rabi frequency $\Omega _{b}$ with the pump Rabi frequency $\Omega _{c}$ as a parameter. For the EIT and EIA regions, the plots show the FWHM of the observed peaks and for the AT case, the plots show the peak separations.
Fig. 4.
Fig. 4. Energy level diagram for $^{85}Rb$ $D_2$ transition. $\Omega _{p}$ , $\Omega _{b}$ , and $\Omega _{c}$ are the probe, the coupling and the pump Rabi frequencies respectively.
Fig. 5.
Fig. 5. Schematic dressed state basis for (a) EIT (b) EIA and (c) AT. Bare states diagram are in the Fig. 4.
Fig. 6.
Fig. 6. Numerical plots for the populations of the different states (mentioned in the legends). (a) The population distribution for $\Omega _{b}=0$ , (b) the population distribution for $\Omega _{b} = 10$ $MHz$ and (c) the population distribution for $\Omega _{b} = 20$ $MHz$ . Here the probe Rabi frequency $\Omega _p$ and the pump Rabi frequency $\Omega _c$ are taken to be $1$ $MHz$ and $10$ $MHz$ respectively for all the simulations.
Fig. 7.
Fig. 7. Numerical simulations of the probe coherence term $\rho _{31}$ (probe absorption) as function of time(t) and pump detunning ( $\Delta _{c}$ ). (a) The EIT case $\Omega _{b}=0$ , (b) The EIA case $\Omega _{b} = 10$ $MHz$ and (c) the AT case $\Omega _{b} = 20$ $MHz$ . Here probe Rabi frequency $\Omega _p$ and pump Rabi frequency $\Omega _c$ are taken to be $1$ $MHz$ and $10$ $MHz$ respectively for all the simulations.
Fig. 8.
Fig. 8. Theoretical simulation of the probe coherence $\rho _{31}$ as a functions of pump detunning $\Delta _{c}$ considering the M-B distribution (a) The EIT case when $\Omega _{b} = 0$ $MHz$ , (b) the EIA case when $\Omega _{b} = 10$ $MHz$ and (c) the AT case when $\Omega _{b} = 20$ $MHz$ . Here the probe Rabi frequency $\Omega _p = 1$ $MHz$ and the pump Rabi frequency $\Omega _c = 10$ $MHz$ are taken for all the simulations.

Equations (15)

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H = [ Δ b + Δ p 0 Ω p Ω b 0 Δ c Ω c 0 Ω p Ω c Δ c Δ p 0 Ω b 0 0 Δ b ]
H = [ Δ b 0 0 Ω b 0 Δ c Ω c 0 0 Ω c Δ c 0 Ω b 0 0 Δ b ]
R = [ 1 0 0 0 0 cos θ sin θ 0 0 sin θ cos θ 0 0 0 0 1 ]
U d = R . U = [ | 1 cos θ | 2 + sin θ | 3 sin θ | 2 + cos θ | 3 | 4 ]
R = [ cos α 0 0 sin α 0 cos θ sin θ 0 0 sin θ cos θ 0 sin α 0 0 cos α ]
U d = R . U = [ cos α | 1 + sin α | 4 cos θ | 2 + sin θ | 3 sin θ | 2 + cos θ | 3 sin α | 1 + cos α | 4 ]
R = [ cos α 0 0 sin α 0 1 0 0 0 0 1 0 sin α 0 0 cos α ]
U d = R . U = [ cos α | 1 + sin α | 4 | 2 | 3 sin α | 1 + cos α | 4 ]
d d t ρ = i [ H , ρ ]
d ρ 11 d t = Γ 31 ρ 33 + Γ 44 ρ 44 Γ 12 ρ 11 + Γ 21 ρ 22 i 2 Ω p ρ 13 + i 2 Ω p ρ 31 i 2 Ω b ρ 14 + i 2 Ω b ρ 41 d ρ 22 d t = Γ 32 ρ 33 + Γ 12 ρ 11 Γ 21 ρ 22 i 2 Ω c ρ 23 + i 2 Ω c ρ 32 d ρ 33 d t = Γ 33 ρ 33 + i 2 Ω p ρ 13 i 2 Ω p ρ 31 + i 2 Ω c ρ 23 i 2 Ω c ρ 32 d ρ 44 d t = Γ 44 ρ 44 + i 2 Ω b ρ 14 i 2 Ω b ρ 41 d ρ 14 d t = D 14 1 ρ 14 + i 2 Ω p ρ 34 + i 2 Ω b ( ρ 44 ρ 11 ) d ρ 23 d t = D 23 1 ρ 23 i 2 Ω p ρ 21 + i 2 Ω c ( ρ 33 ρ 22 ) d ρ 21 d t = D 21 1 ρ 21 i 2 Ω p ρ 23 i 2 Ω b ρ 24 + i 2 Ω c ρ 31 d ρ 34 d t = D 34 1 ρ 34 + i 2 Ω p ρ 14 + i 2 Ω c ρ 24 i 2 Ω b ρ 31 d ρ 24 d t = D 24 1 ρ 24 + i 2 Ω c ρ 34 i 2 Ω b ρ 21 d ρ 31 d t = D 31 1 ρ 31 i 2 Ω p ( ρ 33 ρ 11 ) i 2 Ω b ρ 34 + i 2 Ω c ρ 21
ρ 31 = i 2 Ω p ( ρ 33 0 ρ 11 0 ) + Ω p Ω b 4 D 34 [ 1 Ω c 2 4 A 24 ( D 34 + D 21 ) ] ρ 14 0 + Ω p Ω c 4 D 21 [ 1 Ω b 2 4 A 24 ( D 34 + D 21 ) ] ρ 23 0 A 31 1 Ω b 2 Ω c 2 16 ( D 34 + D 21 ) 2 A 24
A 31 1 = D 31 1 + Ω c 2 4 D 21 + Ω b 2 4 D 34 A 24 1 = D 24 1 + Ω c 2 4 D 34 + Ω b 2 4 D 21 ρ 14 0 = i 2 D 14 Ω b ( ρ 44 0 ρ 11 0 ) ρ 23 0 = i 2 D 23 Ω c ( ρ 33 0 ρ 22 0 )
ρ 22 0 = Γ 12 2 × [ Ω c 2 + Γ 2 + 4 ( Δ c + k v ) 2 ] [ Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 ] Γ 21 2 [ Ω c 2 + Γ 2 + 4 ( Δ c + k v ) 2 ] [ 2 Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 ] + Γ 12 2 [ 2 Ω c 2 + Γ 2 + 4 ( Δ c + k v ) 2 ] [ Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 ] + Ω c 2 Γ 4 [ 2 Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 ] ρ 33 0 = Ω c 2 Ω c 2 + ( Γ 2 + 4 ( Δ c + k v ) 2 ) × ρ 22 0 ρ 11 0 = Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 2 Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 2 Ω c 2 + Γ 2 + 4 ( Δ c + k v ) 2 Ω c 2 + Γ 2 + 4 ( Δ c + k v ) 2 × Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 2 Ω b 2 + Γ 2 + 4 ( Δ b + k v ) 2 × ρ 22 0 ρ 44 0 = Ω b 2 Ω b 2 + ( Γ 2 + 4 ( Δ b + k v ) 2 ) × ρ 11 0
χ = μ ϵ 0 E p N ( k v ) ρ 31 d ( k v )
N ( k v ) = N 0 π k 2 u 2 e ( k v ) 2 / ( k u ) 2

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