Abstract

We explore the alteration that occurs in Autler–Townes (AT) splitting caused by the competition of phonon intensity and polarized dressing fields in second-order fluorescence (FL) signals in both the spectral and the time domain. By varying the polarization dressing along with temperature, we control the AT-splitting of the FL signal. Furthermore, we analyze the noise correlation and squeezing of spontaneous four-wave mixing (SP-FWM) to investigate the effect of phonon and polarized dressing fields. These results are applied for optical temperature sensors and switching devices, which can be controlled by the competition between phonon and polarization dressing. Optical temperature sensor sensitivity and switching speeds are 96% and 15 ns, respectively.

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

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References

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  1. P. B. Chapple, K. G. H. Baldwin, and H. A. Bachor, “Interference between competing quantum-mechanical pathways for fourwave mixing,” J. Opt. Soc. Am. B 6(2), 180 (1989).
    [Crossref]
  2. W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
    [Crossref]
  3. S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
    [Crossref]
  4. S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
    [Crossref]
  5. A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
    [Crossref]
  6. V. Milner and Y. Prior, “Multilevel dark states: Coherent population trapping with elliptically polarized incoherent light,” Phys. Rev. Lett. 80(5), 940–943 (1998).
    [Crossref]
  7. B. Wang, Y. J. Xiao, X. Yang, C. Xie, H. Wang, and M. Xiao, “Multi-dark-state resonances in cold multi-Zeeman-sublevel atoms,” Opt. Lett. 31(24), 3647–3649 (2006).
    [Crossref]
  8. S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
    [Crossref]
  9. Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
    [Crossref]

2007 (2)

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

2006 (2)

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

B. Wang, Y. J. Xiao, X. Yang, C. Xie, H. Wang, and M. Xiao, “Multi-dark-state resonances in cold multi-Zeeman-sublevel atoms,” Opt. Lett. 31(24), 3647–3649 (2006).
[Crossref]

2001 (1)

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

2000 (1)

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

1998 (2)

V. Milner and Y. Prior, “Multilevel dark states: Coherent population trapping with elliptically polarized incoherent light,” Phys. Rev. Lett. 80(5), 940–943 (1998).
[Crossref]

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

1989 (1)

Akulshin, A. M.

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

Bachor, H. A.

Baldwin, K. G. H.

Barreiro, S.

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

Borisov, R. S.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Chapple, P. B.

Du, S. W.

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

Han, Y. X.

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Lezama, A.

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

Li, S. J.

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Lipsich, A.

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

Magno, W. C.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

Malyukin, Yu. V.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Manykin, E. A.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Marchenko, D. V.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Milner, V.

V. Milner and Y. Prior, “Multilevel dark states: Coherent population trapping with elliptically polarized incoherent light,” Phys. Rev. Lett. 80(5), 940–943 (1998).
[Crossref]

Minkov, B. I.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Nussenzveig, P.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

Oh, E.

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

Peng, K. C.

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Petrenko, E. A.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Prandini, R. B.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

Prior, Y.

V. Milner and Y. Prior, “Multilevel dark states: Coherent population trapping with elliptically polarized incoherent light,” Phys. Rev. Lett. 80(5), 940–943 (1998).
[Crossref]

Rubin, M. H.

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

Seminozhenko, V. P.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Vianna, S. S.

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

Wang, B.

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

B. Wang, Y. J. Xiao, X. Yang, C. Xie, H. Wang, and M. Xiao, “Multi-dark-state resonances in cold multi-Zeeman-sublevel atoms,” Opt. Lett. 31(24), 3647–3649 (2006).
[Crossref]

Wang, H.

B. Wang, Y. J. Xiao, X. Yang, C. Xie, H. Wang, and M. Xiao, “Multi-dark-state resonances in cold multi-Zeeman-sublevel atoms,” Opt. Lett. 31(24), 3647–3649 (2006).
[Crossref]

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Wen, J. M.

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

Xiao, M.

B. Wang, Y. J. Xiao, X. Yang, C. Xie, H. Wang, and M. Xiao, “Multi-dark-state resonances in cold multi-Zeeman-sublevel atoms,” Opt. Lett. 31(24), 3647–3649 (2006).
[Crossref]

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Xiao, Y. J.

Xie, C.

Yang, X.

Yang, X. D.

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Yin, G. Y.

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

Znamenskii, N. V.

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

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

Low Temp. Phys. (1)

Yu. V. Malyukin, B. I. Minkov, R. S. Borisov, V. P. Seminozhenko, N. V. Znamenskii, E. A. Manykin, D. V. Marchenko, and E. A. Petrenko, “Low-temperature spectroscopy of nonequivalent Pr3+ optical centers in a Y2SiO5 crystal,” Low Temp. Phys. 24(6), 432–436 (1998).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (2)

W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: observation of interference fringes,” Phys. Rev. A 63(6), 063406 (2001).
[Crossref]

S. W. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: interference and entanglement,” Phys. Rev. A 76(1), 013803 (2007).
[Crossref]

Phys. Rev. A: At., Mol., Opt. Phys. (2)

A. Lipsich, S. Barreiro, A. M. Akulshin, and A. Lezama, “Absorption spectra of driven degenerate two-level atomic systems,” Phys. Rev. A: At., Mol., Opt. Phys. 61(5), 053803 (2000).
[Crossref]

S. J. Li, B. Wang, X. D. Yang, Y. X. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A: At., Mol., Opt. Phys. 74(3), 033821 (2006).
[Crossref]

Phys. Rev. Lett. (2)

V. Milner and Y. Prior, “Multilevel dark states: Coherent population trapping with elliptically polarized incoherent light,” Phys. Rev. Lett. 80(5), 940–943 (1998).
[Crossref]

S. W. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98(5), 053601 (2007).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. (a) Simplified energy levels diagram of Pr3+ ions in YSO crystal. (b) Energy levels of two-mode system in Pr3+: YSO and the laser coupling configuration. (c) Experimental setup scheme, where PMT: photomultiplier tube, QWP: quarter-wave plate, PBS: polarized beam splitter, E S is the Stokes signal and E AS is the anti-Stokes signal. (d) Zeeman energy-levels and transition paths at different polarization states, where solid arrows shows Transitions for the linearly polarized beam and Dotted arrows shows transitions for the right circularly polarized beam. (e) Schematic image of theta angle.
Fig. 2.
Fig. 2. (a) Intensity of second-order FL spectral signal obtained by scanning E 1 and changing E 1 polarization by using QWP in two-level system at temperature of 120 K. (b) Corresponding Time domain FL signal of (a). (c-e) are the intensities of second-order FL spectral signal by changing power of E 1 from high (7 mW) to low (1 mW) for linear, right elliptical and right circular polarizations, respectively at temperature of 120 K. (f-h) show the corresponding time domain signal intensities of (c-e) respectively.
Fig. 3.
Fig. 3. (a-c) Intensity of FL spectral signal at different temperature for linear, right elliptical and right circular polarization, respectively. (d-g) Time domain intensity signal at different polarization for different temperature 77 K, 110 K, 160 K, and 300 K respectively. (h-j) Schematic image of dressing sensitivity versus temperature for linear, elliptical and circular polarization respectively.
Fig. 4.
Fig. 4. (a1-a5) shows intensity of FL spectral signal at various polarization angles 00, 100, 22.50, 320 and 450 respectively. (b1-b3) shows intensity of FL spectral signal at polarization angle −450, 00 and 450 respectively.
Fig. 5.
Fig. 5. (a1-a3) and (b1-b3) Two-mode intensity noise correlation between the output signals E S/AS and FL signal in two level by changing polarization of E 1 at 77 K and 300 K, respectively.
Fig. 6.
Fig. 6. (a1-a3) Three-mode intensity noise correlation between signals E S , E AS and FL signal in two level by changing polarization of E 1 at 110k. (b1-b3) shows corresponding three mode squeezing of (a1)-(a3), respectively.

Equations (15)

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χ x = ( χ x y x ( 2 ) + χ y x x ( 2 ) ) cos 4 θ + sin 4 θ 2 | sin 2 θ cos 2 θ |
χ y = ( χ x x y ( 2 ) ( cos 4 θ + sin 4 θ ) + 2 χ y y y ( 2 ) ) sin 2 θ cos 2 θ
ρ 1 M 1 M ( 2 ) = | G 1 M | 2 ( Γ 1 M 0 M + i Δ 1 + | G 1 M | 2 / Γ 0 M 0 M ) ( Γ 1 M 1 M + | G 1 M | 2 / ( Γ 1 M 0 M + i Δ 1 ) )
ρ 1 M + 1 1 M + 1 ( 2 ) = | G 1 + | 2 ( Γ 1 M + 1 0 M + i Δ 1 + | G 1 M + | 2 / Γ 0 M 0 M ) ( Γ 1 M + 1 1 M + 1 + | G 1 M + | 2 / ( Γ 1 M + 1 0 M + i Δ 1 ) )
ρ 0 M 0 M ( 0 ) G P M 0 ρ 1 M 0 M ( 1 ) G S M 0 ρ 0 M 0 M ( 2 ) G P M 0 ρ 1 M 0 M ( A S ) ( 3 ) ρ 0 M 0 M ( 0 ) G P M 0 ρ 1 M 0 M ( 1 ) G A S M 0 ρ 0 M 0 M ( 2 ) G P M 0 ρ 1 M 0 M ( S ) ( 3 )
ρ A S ( 3 ) = M = ± 1 2 , ± 3 2 , ± 5 2 i G p 0 ( Γ 1 M 0 M + i Δ p + | G p 0 | 2 / Γ 0 M 0 M ) i G S 0 G p 0 Γ 0 M 0 M ( Γ 1 M 0 M + i Δ p | G p 0 | 2 | G p 0 | 2 / Γ 0 M 0 M )
ρ S ( 3 ) = M = ± 1 2 , ± 3 2 , ± 5 2 i G p 0 ( Γ 1 M 0 M + i Δ p + | G p 0 | 2 / Γ 0 M 0 M ) G A S 0 G p 0 Γ 0 M 0 M ( Γ 1 M 0 M + i Δ p | G p 0 | 2 / Γ 0 M 0 M )
ρ 0 M 0 M ( 0 ) G P M 0 ρ 1 M 0 M ( 1 ) G S M + 1 0 ρ 0 M 0 M ( 2 ) G P M + 1 0 ρ 1 M 0 M ( A S ) ( 3 ) ρ 0 M 0 M ( 0 ) G P M 0 ρ 1 M 0 M ( 1 ) G A S M 1 0 ρ 0 M 0 M ( 2 ) G P M 1 0 ρ 1 M 10 M ( S ) ( 3 )
ρ A S ( 3 ) = M = ± 1 2 , ± 3 2 , ± 5 2 i G P M 0 ( Γ 1 M 0 M + i Δ p + | G p 0 | 2 / Γ 0 M + 1 0 M ) G S M + 1 0 G p M + 1 0 Γ 0 M + 1 0 M ( Γ 1 M + 1 0 M + i Δ p + | G p 0 | 2 / Γ 0 M + 1 0 M )
ρ S ( 3 ) = M = ± 1 2 , ± 3 2 , ± 5 2 i G P 0 ( Γ 1 M 0 M + i Δ p + | G p 0 | 2 / Γ 0 M 0 M ) G A S M 1 0 G p 0 Γ 0 M 1 0 M ( Γ 1 M 1 0 M + i Δ p + | G p 0 | 2 / Γ 0 M 1 0 M )
G i j ( 2 ) ( τ ) = δ I ^ i ( t i ) δ I ^ j ( t j ) δ I ^ S ( t S ) 2 δ I ^ AS ( t AS ) 2
A s / a s = R 1 | A 1 | 2 [ e 2 Γ S L | τ | + e 2 Γ A S L | τ | 2 cos ( Ω e | τ | ) e ( Γ S L + Γ A S L ) | τ | ]
S q ( 2 ) = L o g 10 δ 2 ( I ^ i I ^ j ) δ 2 ( I ^ i + I ^ j )
G i j k ( 3 ) ( τ ) = δ I ^ i ( t i ) δ I ^ j ( t j ) δ I ^ k ( t k ) δ I ^ i ( t i ) 2 δ I ^ j ( t j ) 2 δ I ^ k ( t k ) 2
S q ( 3 ) = L o g 10 δ 2 ( I ^ i I ^ j I ^ k ) δ 2 ( I ^ i + I ^ j + I ^ k )

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