Abstract

We theoretically investigate temporal dynamics of the second order cross correlation function at zero delay time ($G^{(2)}_{12}(t)$) and spectral entanglement of two photons emitted from an atomic three-level cascade. In Heisenberg’s picture, a closed set of quantum kinetic equations of motion for $G^{(2)}_{12}(t)$ is derived within density matrix formalism with cluster expansion rule. $G^{(2)}_{12}(t)$ shows qualitatively distinctive features depending on the spectral entanglement of two photons. Although incoherent photon pairs generated from spontaneous radiation of the excited electron are not entangled, their correlation and anti-correlation properties can be found in $G^{(2)}_{12}(t)$ depending on the radiative decay rates. In the coherent excitation regime where the light emitter is located in a high Q-cavity, and its atomic polarizations are predominantly initialized, spectral entanglement between two coherent photons is established. We show that $G^{(2)}_{12}(t)$ is well fitted by the entanglement criterion by Duan-Giedke-Cirac-Zoller and explain the close relationship between them by means of the optically forbidden transition in the three-level cascade.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. d. Silva, Y. Yu, J. Iles-Smith, K. Srinivasan, A. Rastelli, J. Li, and X. Wang, “A solid-state source of strongly entangled photon pairs with high brightness and indistinguishability,” Nat. Nanotechnol. 14(6), 586–593 (2019).
[Crossref]

2018 (3)

C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, “Strong coupling between photons of two light fields mediated by one atom,” Nat. Phys. 14(9), 885–889 (2018).
[Crossref]

F. Ripka, H. Kübler, R. Löw, and T. Pfau, “A room-temperature single-photon source based on strongly interacting rydberg atoms,” Science 362(6413), 446–449 (2018).
[Crossref]

J. Park, T. Jeong, H. Kim, and H. S. Moon, “Time-energy entangled photon pairs from doppler-broadened atomic ensemble via collective two-photon coherence,” Phys. Rev. Lett. 121(26), 263601 (2018).
[Crossref]

2017 (6)

K. J. Ahn, “Resonance fluorescence of ladder- and triangular-type three-level systems: Continuous coherent photon generation,” J. Korean Phys. Soc. 71(10), 657–664 (2017).
[Crossref]

S. Bounouar, M. Strauß, A. Carmele, P. Schnauber, A. Thoma, M. Gschrey, J.-H. Schulze, A. Strittmatter, S. Rodt, A. Knorr, and S. Reitzenstein, “Path-controlled time reordering of paired photons in a dressed three-level cascade,” Phys. Rev. Lett. 118(23), 233601 (2017).
[Crossref]

S. Gasparinetti, M. Pechal, J.-C. Besse, M. Mondal, C. Eichler, and A. Wallraff, “Correlations and entanglement of microwave photons emitted in a cascade decay,” Phys. Rev. Lett. 119(14), 140504 (2017).
[Crossref]

M.-X. Dong, W. Zhang, S. Shi, K. Wang, Z.-Y. Zhou, S.-L. Liu, D.-S. Ding, and B.-S. Shi, “Two-color hyper-entangled photon pairs generation in a cold 85rb atomic ensemble,” Opt. Express 25(9), 10145–10152 (2017).
[Crossref]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator,” Science 358(6360), 203–206 (2017).
[Crossref]

A. Zeilinger, “Light for the quantum. Entangled photons and their applications: a very personal perspective,” Phys. Scr. 92(7), 072501 (2017).
[Crossref]

2016 (3)

G. Popkin, “Quest for qubits,” Science 354(6316), 1090–1093 (2016).
[Crossref]

Y.-M. He, O. Iff, N. Lundt, V. Baumann, M. Davanco, K. Srinivasan, S. Höfling, and C. Schneider, “Cascaded emission of single photons from the biexciton in monolayered WSe2,” Nat. Commun. 7(1), 13409 (2016).
[Crossref]

Y.-S. Lee, S. M. Lee, H. Kim, and H. S. Moon, “Highly bright photon-pair generation in doppler-broadened ladder-type atomic system,” Opt. Express 24(24), 28083–28091 (2016).
[Crossref]

2015 (1)

Z. H. Peng, Y.-x. Liu, J. T. Peltonen, T. Yamamoto, J. S. Tsai, and O. Astafiev, “Correlated emission lasing in harmonic oscillators coupled via a single three-level artificial atom,” Phys. Rev. Lett. 115(22), 223603 (2015).
[Crossref]

2013 (1)

B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble,” Phys. Rev. Lett. 111(12), 123602 (2013).
[Crossref]

2012 (5)

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Generation of non-classical correlated photon pairs via a ladder-type atomic configuration: theory and experiment,” Opt. Express 20(10), 11433–11444 (2012).
[Crossref]

A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from the mollow triplet sidebands of a quantum dot,” Nat. Photonics 6(4), 238–242 (2012).
[Crossref]

K. V. Kheruntsyan, J.-C. Jaskula, P. Deuar, M. Bonneau, G. B. Partridge, J. Ruaudel, R. Lopes, D. Boiron, and C. I. Westbrook, “Violation of the cauchy-schwarz inequality with matter waves,” Phys. Rev. Lett. 108(26), 260401 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

H.-R. Noh and H. S. Moon, “Transmittance signal in real ladder-type atoms,” Phys. Rev. A 85(3), 033817 (2012).
[Crossref]

2010 (2)

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6(11), 894–899 (2010).
[Crossref]

M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, “Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution,” Opt. Express 18(19), 20229–20241 (2010).
[Crossref]

2008 (4)

C. H. R. Ooi, “Effects of chirped laser pulses on nonclassical correlations and entanglement of photon pairs,” Phys. Rev. A 77(6), 063805 (2008).
[Crossref]

E. A. Sete, “Violation of classical inequalities and EPR correlations in a two-mode three-level atomic system,” Int. J. Quantum Inf. 06(04), 885–898 (2008).
[Crossref]

J. E. Avron, G. Bisker, D. Gershoni, N. H. Lindner, E. A. Meirom, and R. J. Warburton, “Entanglement on demand through time reordering,” Phys. Rev. Lett. 100(12), 120501 (2008).
[Crossref]

M. Kira and S. W. Koch, “Cluster-expansion representation in quantum optics,” Phys. Rev. A 78(2), 022102 (2008).
[Crossref]

2006 (3)

R. J. Young, R. M. Stevenson, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “Improved fidelity of triggered entangled photons from single quantum dots,” New J. Phys. 8(2), 29 (2006).
[Crossref]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96(9), 093604 (2006).
[Crossref]

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

2005 (3)

K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B 71(15), 153309 (2005).
[Crossref]

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71(2), 022501 (2005).
[Crossref]

H. Xiong, M. O. Scully, and M. S. Zubairy, “Correlated spontaneous emission laser as an entanglement amplifier,” Phys. Rev. Lett. 94(2), 023601 (2005).
[Crossref]

2001 (1)

E. Moreau, I. Robert, L. Manin, V. Thierry-Mieg, J. M. Gérard, and I. Abram, “Quantum cascade of photons in semiconductor quantum dots,” Phys. Rev. Lett. 87(18), 183601 (2001).
[Crossref]

2000 (2)

O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot,” Phys. Rev. Lett. 84(11), 2513–2516 (2000).
[Crossref]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

1999 (1)

M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The fermionic hanbury brown and twiss experiment,” Science 284(5412), 296–298 (1999).
[Crossref]

1996 (1)

1986 (2)

D. T. Pegg, R. Loudon, and P. L. Knight, “Correlations in light emitted by three-level atoms,” Phys. Rev. A 33(6), 4085–4091 (1986).
[Crossref]

M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34(2), 1260–1276 (1986).
[Crossref]

1985 (3)

A. Al-Hilfy and R. Loudon, “Theory of photon correlations in two-photon cascade emission,” J. Phys. B: At. Mol. Phys. 18(18), 3697–3712 (1985).
[Crossref]

H. J. Kimble, A. Mezzacappa, and P. W. Milonni, “Time dependence of photon correlations in a three-level atomic cascade,” Phys. Rev. A 31(6), 3686–3697 (1985).
[Crossref]

D. Meschede, H. Walther, and G. Müller, “One-atom maser,” Phys. Rev. Lett. 54(6), 551–554 (1985).
[Crossref]

1980 (1)

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43(7), 913–949 (1980).
[Crossref]

1967 (1)

G. H. Nussbaum and F. M. Pipkin, “Correlation of photons in cascade and the coherence time of the 63p1 state of mercury,” Phys. Rev. Lett. 19(19), 1089–1092 (1967).
[Crossref]

Abram, I.

E. Moreau, I. Robert, L. Manin, V. Thierry-Mieg, J. M. Gérard, and I. Abram, “Quantum cascade of photons in semiconductor quantum dots,” Phys. Rev. Lett. 87(18), 183601 (2001).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, Quantum optics (Cambridge Univ. Press, 2013).

Ahn, K. J.

K. J. Ahn, “Resonance fluorescence of ladder- and triangular-type three-level systems: Continuous coherent photon generation,” J. Korean Phys. Soc. 71(10), 657–664 (2017).
[Crossref]

K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B 71(15), 153309 (2005).
[Crossref]

Akopian, N.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Al-Hilfy, A.

A. Al-Hilfy and R. Loudon, “Theory of photon correlations in two-photon cascade emission,” J. Phys. B: At. Mol. Phys. 18(18), 3697–3712 (1985).
[Crossref]

Aspelmeyer, M.

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator,” Science 358(6360), 203–206 (2017).
[Crossref]

Aßmann, M.

Astafiev, O.

Z. H. Peng, Y.-x. Liu, J. T. Peltonen, T. Yamamoto, J. S. Tsai, and O. Astafiev, “Correlated emission lasing in harmonic oscillators coupled via a single three-level artificial atom,” Phys. Rev. Lett. 115(22), 223603 (2015).
[Crossref]

Atkinson, P.

R. J. Young, R. M. Stevenson, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “Improved fidelity of triggered entangled photons from single quantum dots,” New J. Phys. 8(2), 29 (2006).
[Crossref]

Avron, J.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Avron, J. E.

J. E. Avron, G. Bisker, D. Gershoni, N. H. Lindner, E. A. Meirom, and R. J. Warburton, “Entanglement on demand through time reordering,” Phys. Rev. Lett. 100(12), 120501 (2008).
[Crossref]

Baumann, V.

Y.-M. He, O. Iff, N. Lundt, V. Baumann, M. Davanco, K. Srinivasan, S. Höfling, and C. Schneider, “Cascaded emission of single photons from the biexciton in monolayered WSe2,” Nat. Commun. 7(1), 13409 (2016).
[Crossref]

Bayer, M.

Benson, O.

O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot,” Phys. Rev. Lett. 84(11), 2513–2516 (2000).
[Crossref]

Berlatzky, Y.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
[Crossref]

Berman, P. R.

P. R. Berman and R. C. O’Connell, “Constraints on dephasing widths and shifts in three-level quantum systems,” Phys. Rev. A 71(2), 022501 (2005).
[Crossref]

Berstermann, T.

Besse, J.-C.

S. Gasparinetti, M. Pechal, J.-C. Besse, M. Mondal, C. Eichler, and A. Wallraff, “Correlations and entanglement of microwave photons emitted in a cascade decay,” Phys. Rev. Lett. 119(14), 140504 (2017).
[Crossref]

Bisker, G.

J. E. Avron, G. Bisker, D. Gershoni, N. H. Lindner, E. A. Meirom, and R. J. Warburton, “Entanglement on demand through time reordering,” Phys. Rev. Lett. 100(12), 120501 (2008).
[Crossref]

Boiron, D.

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Nat. Nanotechnol. (1)

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Nat. Photonics (1)

A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from the mollow triplet sidebands of a quantum dot,” Nat. Photonics 6(4), 238–242 (2012).
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Nat. Phys. (2)

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C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, “Strong coupling between photons of two light fields mediated by one atom,” Nat. Phys. 14(9), 885–889 (2018).
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New J. Phys. (1)

R. J. Young, R. M. Stevenson, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “Improved fidelity of triggered entangled photons from single quantum dots,” New J. Phys. 8(2), 29 (2006).
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Opt. Express (4)

Opt. Lett. (1)

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Figures (8)

Fig. 1.
Fig. 1. Energy level configuration of an atomic TLC. The electronic transitions between $|0\rangle \,\leftrightarrow \,|1\rangle$ and $|1\rangle \,\leftrightarrow \,|2\rangle$ are induced by two Rabi frequencies $\Omega _1$ and $\Omega _2$ at center frequencies $\omega _{L_1}$ and $\omega _{L_2}$, respectively. The direct optical transition between $|0\rangle \,\leftrightarrow \,|2\rangle$ is forbidden. The photon number densities $\langle c^\dagger _{\textbf {q}} c_{\textbf {q}} \rangle$ and $\langle c^\dagger _{\textbf {k}} c_{\textbf {k}} \rangle$ are produced by $|0\rangle \,\leftrightarrow \,|1\rangle$ and $|1\rangle \,\leftrightarrow \,|2\rangle$ transitions, respectively. The cross correlation function at zero delay time is defined as $G^{(2)}_{12}(t)=\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle$.
Fig. 2.
Fig. 2. (a) Spontaneous two-photon radiation from the successive transition of the electron initially at the state $|2\rangle$. (b) The same initial conditions can be experimentally prepared by two ultrashort 1.1$\pi$-Gaussian pulses as demonstrated by electronic population densities $(\Gamma _{10}=\Gamma _0)$.
Fig. 3.
Fig. 3. The analytically solved $g^{(2)}_{12}(t)$ (ASM) is compared with the numerical results obtained by two ultrashort $1.1\,\pi$-pulses excitation (PEM) and the spontaneous radiative decay of the electron initially at $|2\rangle$ (SDM) for two cases. (a) $\Gamma _{10}=10\,\Gamma _{21} (\Gamma _{10}=\Gamma _0)$. (b) $\Gamma _{21}=10\,\Gamma _{10} (\Gamma _{21}=\Gamma _0)$. The inset in each figure shows the maximum of $g^{(2)}_{12}(t)$ calculated by the PEM.
Fig. 4.
Fig. 4. Temporal evolution of normalized photon correlation functions $\bar {G}^{(2)}_{12}={G}^{(2)}_{12}/(4|g_{01}(q)|^2|g_{12}(k)|^2)$, $\bar {n}_k=n_k/(2|g_{12}(k)|^2)$, and $\bar {n}_q=n_q/(2|g_{01}(q)|^2)$ for $\Gamma _{10}>\Gamma _{21}$ (red solid lines) and $\Gamma _{10}<\Gamma _{21}$ (blue dashed lines).
Fig. 5.
Fig. 5. (a) Transition between correlation and anti-correlation for four different values of the ratio $\Gamma _{21}/\Gamma _{10} (\Gamma _{10}=\Gamma _{0})$. (b) The convergence of anti-correlation when the ratio $\Gamma _{10}/\Gamma _{21} (\Gamma _{21}=\Gamma _{0})$ is reduced below 0.01.
Fig. 6.
Fig. 6. (a) Coherent two-photon generation by using a short and weak Gaussian pulse $(\int \Omega _1(t)dt=0.01\pi )$ followed by a strong one $(\int \Omega _2(t)dt=\pi )$ in a doubly resonant cavity. (b) Temporal dynamics of the induced atomic polarizations and electron population densities (inset) as a function of the normalized time.
Fig. 7.
Fig. 7. (a) Entanglement $D(t)$ obtained by the numerically solved result (NS) is compared with the analytical solution (AS). (b) Comparison of the correlation functions of photons $|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ multiplied by a constant $C_0=0.57$ with $G_{12}^{(2)}(t)$ for $Q=8\times 10^8\,(\kappa =1.51\,\textrm {MHz},\Gamma _c=5\kappa ,\,g=2\Gamma _c)$.
Fig. 8.
Fig. 8. (a) $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ for three different values of (a) the Q factor and (b) $\Gamma _c$ with respect to $\kappa$. Inset in each figure shows the corresponding $G^{(2)}_{12}(t)$. While $\Gamma _c=5\kappa$ and $g=2\Gamma _c$ are maintained for different values of $Q\,(\kappa )$ for (a), $Q=8\times 10^8\,(\kappa =1.51\,\textrm {MHz})$ is fixed to varying $\Gamma _c$ and $g=2\Gamma _c$ for (b).

Equations (48)

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H ^ = H ^ 0 + H ^ l + H ^ p ,
H ^ 0 = λ ω λ a λ a λ + l ν l c l c l ,
H ^ l = Ω ~ 1 ( t ) ( a 0 a 1 + a 1 a 0 ) Ω ~ 2 ( t ) ( a 1 a 2 + a 2 a 1 ) ,
H ^ p = i λ , μ l g λ μ ( l ) a λ a μ ( c l c l ) ,
D ( t ) = ( Δ u ^ ) 2 + ( Δ v ^ ) 2 = 2 { 1 + c k c k | c k | 2 + c q c q | c q | 2 + 2 R e [ c k c q c k c q ] } ,
c k c k = c k c k c + c k c k = c k c k c + | c k | 2 ,
c k c q c q c k = c k c q c q c k c + 12 | c k | 2 | c q | 2 + c k c k c q c q + | c k c q | 2 + | c k c q | 2 3 { c k c k | c q | 2 + c q c q | c k | 2 + 2 R e [ c k c q c q c k + c k c q c k c q ] } + 2 R e [ c k c q c q c k + c q c k c q c q ] ,
n ˙ q 2 | g 01 ( q ) | 2 { 1 10 Γ 0 ( e Γ 10 t + e Γ 21 t 2 e γ 1 t )  for  Γ 10 = Γ 0 = 10 Γ 21 10 Γ 0 ( e γ 1 t e Γ 10 t )  for  Γ 21 = Γ 0 = 10 Γ 10 .
D ( t ) 2 ( 1 + 2 R e [ c k c q ] ) ,
c k c q c q c k c k c q c q c k c + | c k c q | 2 .
t c k c q c q c k = 2 ( κ q + κ k ) c k c q c q c k + 2 R e [ g 01 ( q ) a 0 a 1 c q c k c k ] ,
t a 0 a 1 c q c k c k = [ i ( ω 10 + ν q ) γ 1 κ q 2 κ k ] a 0 a 1 c q c k c k + g 01 ( q ) a 1 a 1 c k c k i Ω 1 ( a 1 a 1 c q c k c k a 0 a 0 c q c k c k ) + i Ω 2 a 0 a 2 c q c k c k ,
t a 1 a 1 c k c k = ( Γ 10 + 2 κ k ) a 1 a 1 c k c k + 2 R e [ g 12 ( k ) a 1 a 2 c k ] i Ω 1 ( a 0 a 1 c k c k a 1 a 0 c k c k ) i Ω 2 ( a 1 a 2 c k c k a 2 a 1 c k c k ) ,
t a 1 a 2 c k = [ i ( ω 21 + ν k ) γ 3 κ k ] a 1 a 2 c k + g 12 ( k ) ρ 2 i Ω 1 a 0 a 2 c k i Ω 2 ( a 2 a 2 c k a 1 a 1 c k ) ,
t a 0 a 1 c q = [ i ( ω 10 + ν q ) γ 1 κ q ] a 0 a 1 c q + g 01 ( q ) ρ 0 i Ω 1 ( a 1 a 1 c q a 0 a 0 c q ) + i Ω 2 a 0 a 2 c q ,
t a 0 a 1 c k = [ i ( ω 10 + ν k ) γ 1 κ k ] a 0 a 1 c k + g 12 ( k ) p 02 , i Ω 1 ( a 1 a 1 c q a 0 a 0 c q ) + i Ω 2 a 0 a 2 c q ,
t a 2 a 1 c q = [ i ( ω 21 ν q ) γ 3 κ q ] a 2 a 1 c q + g 01 ( q ) p 20 , + i Ω 1 a 2 a 0 c q + i Ω 2 ( a 2 a 2 c q a 1 a 1 c q ) ,
t c k c k = 2 κ k c k c k + 2 R e [ g 12 ( k ) a 1 a 2 c k ] ,
t c q c q = 2 κ q c q c q + 2 R e [ g 01 ( q ) a 0 a 1 c q ] ,
t c k c q = [ i ( ν k + ν q ) κ k κ q ] c k c q + g 01 ( q ) a 0 a 1 c k + g 12 ( k ) a 2 a 1 c q ,
t c k = ( i ν k κ k ) c k + g 12 ( k ) p 12 ,
t c q = ( i ν q κ q ) c q + g 01 ( q ) p 01 ,
t p 12 = i ω 21 p 12 + k g 12 ( k ) ( a 2 a 2 c k + a 1 a 1 c k a 2 a 2 c k a 1 a 1 c k ) + q g 01 ( q ) ( a 0 a 2 c q a 0 a 2 c q ) ,
t a 2 a 2 c k = i ν k a 2 a 2 c k + k' g 12 ( k ) ( a 1 a 2 c k' c k a 1 a 2 c k' c k a 2 a 1 c k' c k + a 2 a 1 c k' c k ) + g 12 ( k ) p 21 = t a 2 a 2 c k ,
t a 1 a 1 c k = i ν k a 1 a 1 c k + k' g 12 ( k ) ( a 2 a 1 c k' c k a 2 a 1 c k' c k a 1 a 2 c k' c k + a 1 a 2 c k' c k ) + g 12 ( k ) p 12 = t a 1 a 1 c k
t a 0 a 2 c q = i ( ω 20 + ν q ) a 0 a 2 c q + q' g 01 ( q ) ( a 1 a 2 c q' c q a 1 a 2 c q' c q ) k' g 12 ( k ) ( a 0 a 1 c k' c q a 0 a 1 c k' c q ) ,
t a 0 a 2 c q = i ( ω 20 ν q ) a 0 a 2 c q + q' g 01 ( q ) ( a 1 a 2 c q c q' a 1 a 2 c q' c q ) + g 01 ( q ) p 12 k' g 12 ( k ) ( a 0 a 1 c q c k' a 0 a 1 c k' c q ) .
t a 2 a 2 c k e = g 12 ( k ) p ~ 21 e i ( ω 21 t + ν k ) t = t a 2 a 2 c k e ,
t a 1 a 1 c k e = g 12 ( k ) p ~ 12 e i ( ν k ω 21 ) t = t a 1 a 1 c k e ,
t a 0 a 2 c q e = g 01 ( q ) p ~ 12 e i ( ω 10 ν q ) t .
t p ~ 12 ( t ) = k | g 12 ( k ) | 2 t d t [ p ~ 21 ( t ) e i ( ω 21 + ν k ) t e i ( ω 21 ν k ) t + p ~ 21 ( t ) e i ( ω 21 ν k ) t e i ( ω 21 + ν k ) t p ~ 12 ( t ) e i ( ω 21 + ν k ) ( t t ) p ~ 12 ( t ) e i ( ω 21 ν k ) ( t t ) ] q | g 01 ( q ) | 2 t p ~ 12 ( t ) e i ( ω 10 ν q ) ( t t ) d t ,
t p ~ 12 ( t ) = { k | g 12 ( k ) | 2 t e i ( ω 21 ν k ) ( t t ) d t + q | g 01 ( q ) | 2 t e i ( ω 10 ν q ) ( t t ) d t } p ~ 12 ( t ) ,
t p ~ 12 ( t ) = γ 3 p ~ 12 ( t ) = 1 2 ( Γ 21 + Γ 10 ) p ~ 12 ( t ) ,
t p ~ 01 ( t ) = γ 1 p ~ 01 ( t ) = 1 2 Γ 10 p ~ 01 ( t ) ,
t p ~ 02 ( t ) = γ 2 p ~ 02 ( t ) = 1 2 Γ 21 p ~ 02 ( t ) .
p ~ ˙ 01 = γ 1 p ~ 01 i Ω 1 ( ρ 1 ρ 0 ) e i Δ ω 1 t + i Ω 2 p ~ 02 e i Δ ω 2 t ,
p ~ ˙ 12 = γ 3 p ~ 12 i Ω 1 p ~ 02 e i Δ ω 1 t i Ω 2 ( ρ 2 ρ 1 ) e i Δ ω 2 t ,
p ~ ˙ 02 = γ 2 p ~ 02 i Ω 1 p ~ 12 e i Δ ω 1 t + i Ω 2 p ~ 01 e i Δ ω 2 t ,
ρ ˙ 2 = Γ 21 ρ 2 i Ω 2 ( p ~ 12 e i Δ ω 2 t p ~ 21 e i Δ ω 2 t ) ,
ρ ˙ 1 = Γ 10 ρ 1 + Γ 21 ρ 2 i Ω 1 ( p ~ 01 e i Δ ω 1 t p ~ 10 e i Δ ω 1 t ) + i Ω 2 ( p ~ 12 e i Δ ω 2 t p ~ 21 e i Δ ω 2 t ) ,
ρ ˙ 0 = Γ 10 ρ 1 + i Ω 1 ( p ~ 01 e i Δ ω 1 t p ~ 10 e i Δ ω 1 t ) ,
ρ 2 ( t ) = ρ 2 ( 0 ) e Γ 21 t ,
ρ 1 ( t ) = { Γ 21 ρ 2 ( 0 ) t e Γ 10 t for  Γ 21 = Γ 10 , Γ 21 Γ 21 Γ 10 ρ 2 ( 0 ) ( e Γ 10 t e Γ 21 t ) for  Γ 21 Γ 10 ,
ρ 0 ( t ) = 1 ρ 2 ( t ) ρ 1 ( t ) ,
c k c q c q c k = 4 | g 01 ( q ) | 2 | g 12 ( k ) | 2 ( γ 3 κ Γ 21 ) 2 ρ 2 ( 0 ) e κ k q t { a 1 ( 1 e ( κ k q Γ 21 ) t Γ 21 κ k q 1 e ( κ k q γ 1 k q ) t γ 1 k q κ k q ) + a 2 ( 1 e ( κ k q Γ 10 k ) t Γ 10 k κ k q 1 e ( κ k q γ 1 k q ) t γ 1 k q κ k q ) + a 3 ( 1 e ( κ k q γ 3 k ) t γ 3 k κ k q 1 e ( κ k q γ 1 k q ) t γ 1 k q κ k q ) } ,
a 1 = Γ 21 γ 3 k ( Γ 21 Γ 10 k ) ( γ 1 k q Γ 21 ) , a 2 = ( Γ 21 γ 3 k ) 2 ( Γ 21 Γ 10 k ) ( Γ 10 k γ 3 k ) ( Γ 10 k γ 1 k q ) , a 3 = Γ 21 γ 3 k ( Γ 10 k γ 3 k ) ( γ 3 k γ 1 k q ) , Γ 10 k = Γ 10 + 2 κ k , γ 1 k q = γ 1 + κ k + 2 κ q , γ 3 k = γ 3 + κ k , a n d κ k q = 2 ( κ k + κ q ) . c k c k = 2 | g 12 ( k ) | 2 γ 3 k Γ 21 ρ 2 ( 0 ) e 2 κ k t { 1 e ( 2 κ k Γ 21 ) t Γ 21 2 κ k 1 e ( 2 κ k γ 3 k ) t γ 3 k 2 κ k } ,
c q c q = 2 | g 01 ( q ) | 2 ρ 1 ( 0 ) e 2 κ q t { 1 γ 1 q Γ 10 ( 1 e ( 2 κ q Γ 10 ) t Γ 10 2 κ q 1 e ( 2 κ q γ 1 q ) t γ 1 q 2 κ k ) 1 γ 1 q Γ 21 ( 1 e ( 2 κ q Γ 21 ) t Γ 21 2 κ q 1 e ( 2 κ q γ 1 q ) t γ 1 q 2 κ k ) }  for  Γ 21 Γ 10 ,
c k c q e = g 01 ( q ) g 12 ( k ) p 02 ( 0 ) e ( κ k + κ q ) t γ 1 γ 2 + κ k { 1 e ( κ q γ 1 ) t γ 1 κ q 1 e ( κ k + κ q γ 2 ) t γ 2 κ k κ q }  for  ω 21 , ω 10 γ 1 , γ 2 , κ k , κ q ,

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