Abstract

We present an efficient scheme for generating the tripartite continuous variable entanglement of microwave radiation in a Δ-type three-level artificial atom placed in a transmission line resonator, simultaneously driven resonantly by two strong electromagnetic fields. We show that it is possible to obtain the full tripartite entanglement with the different frequencies in the presence of atom and cavity decays. In our scheme, interestingly, the nonlinear process of simultaneous parametric downconversion and upconversion interactions between microwave modes can be achieved via quantum interference induced by two classical fields and can be greatly enhanced under the condition of resonant driving of the two classical fields. Such a nonlinear interaction is responsible for the generation of the strong tripartite entanglement among three microwave modes. In practice, the system considered here could offer an alternative for implementing the scalable information processing in the solid-state system.

© 2013 Optical Society of America

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  28. F. O. Prado, N. G. de Almeida, M. H. Y. Moussa, and C. J. Villas-Bôas, “Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics,” Phys. Rev. A 73, 043803 (2006).
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  29. X. M. Hu and X. Li, “Quantum interference in enhanced parametric interactions,” J. Phys. B 43, 055502 (2010).
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    [CrossRef]
  31. Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004).
    [CrossRef]
  32. X. H. Yang, Y. Y. Zhou, and M. Xiao, “Generation of multipartite continuous-variable entanglement via atomic spin wave,” Phys. Rev. A 85, 052307 (2012).
    [CrossRef]
  33. A. A. Valido, L. A. Correa, and D. Alonso, “Gaussian tripartite entanglement out of equilibrium,” Phys. Rev. A 88, 012309 (2013).
    [CrossRef]
  34. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
    [CrossRef]
  35. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
    [CrossRef]
  36. O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
    [CrossRef]
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    [CrossRef]
  38. J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically induced transparency with amplification in superconducting circuits,” Phys. Rev. Lett. 105, 073601 (2010).
    [CrossRef]
  39. W. Z. Jia, L. F. Wei, and Z. D. Wang, “Tunable one-dimensional microwave emissions from cyclic-transition three-level artificial atoms,” Phys. Rev. A 83, 023811 (2011).
    [CrossRef]
  40. P. M. Leung and B. C. Sanders, “Coherent control of microwave pulse storage in superconducting circuits,” Phys. Rev. Lett. 109, 253603 (2012).
    [CrossRef]
  41. S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett 94, 100502 (2005).
    [CrossRef]
  42. L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
    [CrossRef]
  43. A. F. Obada, H. A. Hessian, A. A. Mohamed, and A. H. Homid, “Implementing discrete quantum Fourier transform via superconducting qubits coupled to a superconducting cavity,” J. Opt. Soc. Am. B 30, 1178–1185 (2013).
    [CrossRef]
  44. C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
    [CrossRef]
  45. D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
    [CrossRef]
  46. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
    [CrossRef]
  47. V. E. Manucharyan, N. A. Masluk, A. Kamal, J. Koch, L. I. Glazman, and M. H. Devoret, “Evidence for coherent quantum phase slips across a Josephson junction array,” Phys. Rev. B 85, 064521 (2012).
    [CrossRef]
  48. H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
    [CrossRef]
  49. M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler–Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
    [CrossRef]
  50. A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmstrøm, J. M. Martinis, and A. N. Cleland, “Planar superconducting resonators with internal quality factors above one million,” Appl. Phys. Lett. 100, 113510 (2012).
    [CrossRef]

2013

C. Weedbrook, “Continuous-variable quantum key distribution with entanglement in the middle,” Phys. Rev. A 87, 022308 (2013).
[CrossRef]

A. A. Valido, L. A. Correa, and D. Alonso, “Gaussian tripartite entanglement out of equilibrium,” Phys. Rev. A 88, 012309 (2013).
[CrossRef]

A. F. Obada, H. A. Hessian, A. A. Mohamed, and A. H. Homid, “Implementing discrete quantum Fourier transform via superconducting qubits coupled to a superconducting cavity,” J. Opt. Soc. Am. B 30, 1178–1185 (2013).
[CrossRef]

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[CrossRef]

2012

P. M. Leung and B. C. Sanders, “Coherent control of microwave pulse storage in superconducting circuits,” Phys. Rev. Lett. 109, 253603 (2012).
[CrossRef]

V. E. Manucharyan, N. A. Masluk, A. Kamal, J. Koch, L. I. Glazman, and M. H. Devoret, “Evidence for coherent quantum phase slips across a Josephson junction array,” Phys. Rev. B 85, 064521 (2012).
[CrossRef]

A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmstrøm, J. M. Martinis, and A. N. Cleland, “Planar superconducting resonators with internal quality factors above one million,” Appl. Phys. Lett. 100, 113510 (2012).
[CrossRef]

X. H. Yang, Y. Y. Zhou, and M. Xiao, “Generation of multipartite continuous-variable entanglement via atomic spin wave,” Phys. Rev. A 85, 052307 (2012).
[CrossRef]

X. Liang, X. M. Hu, and C. He, “Creating multimode squeezed states and Greenberger–Horne–Zeilinger entangled states using atomic coherent effects,” Phys. Rev. A 85, 032329 (2012).
[CrossRef]

Y. Gu, G. Q. He, and X. F. Wu, “Generation of six-partite continuous-variable entanglement using a nonlinear photonic crystal by frequency conversions,” Phys. Rev. A 85, 052328 (2012).
[CrossRef]

2011

Y. B. Yu, J. T. Sheng, and M. Xiao, “Generation of bright quadricolor continuous-variable entanglement by four-wave-mixing process,” Phys. Rev. A 83, 012321 (2011).
[CrossRef]

C. Y. Zhao, W. H. Tan, J. R. Xu, and F. Ge, “Multipartite continuous-variable entanglement in nondegenerate optical parametric amplification system,” J. Opt. Soc. Am. B 28, 1067–1076 (2011).
[CrossRef]

N. C. Menicucci, “Temporal-mode continuous-variable cluster states using linear optics,” Phys. Rev. A 83, 062314 (2011).
[CrossRef]

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[CrossRef]

W. Z. Jia, L. F. Wei, and Z. D. Wang, “Tunable one-dimensional microwave emissions from cyclic-transition three-level artificial atoms,” Phys. Rev. A 83, 023811 (2011).
[CrossRef]

2010

L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
[CrossRef]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[CrossRef]

W. R. Kelly, Z. Dutton, J. Schlafer, B. Mookerji, T. A. Ohki, J. S. Kline, and D. P. Pappas, “Direct observation of coherent population trapping in a superconducting artificial atom,” Phys. Rev. Lett. 104, 163601 (2010).
[CrossRef]

J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically induced transparency with amplification in superconducting circuits,” Phys. Rev. Lett. 105, 073601 (2010).
[CrossRef]

J. Guo, Z. Zhai, and J. Gao, “Bright quadripartite continuous variable entanglement from coupled intracavity nonlinearities,” J. Opt. Soc. Am. B 27, 518–523 (2010).
[CrossRef]

X. M. Hu and X. Li, “Quantum interference in enhanced parametric interactions,” J. Phys. B 43, 055502 (2010).
[CrossRef]

2009

X. Y. Lü, P. Huang, W. X. Yang, and X. X. Yang, “Entanglement via atomic coherence induced by two strong classical fields,” Phys. Rev. A 80, 032305 (2009).
[CrossRef]

M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler–Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
[CrossRef]

2008

S. Qamar, F. Ghafoor, M. Hillery, and M. S. Zubairy, “Quantum beat laser as a source of entangled radiation,” Phys. Rev. A 77, 062308 (2008).
[CrossRef]

X. M. Hu and J. H. Zou, “Quantum-beat lasers as bright sources of entangled sub-Poissonian light,” Phys. Rev. A 78, 045801 (2008).
[CrossRef]

C. J. McKinstrie, S. J. van Enk, M. G. Raymer, and S. Radic, “Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber,” Opt. Express 16, 2720–2739 (2008).
[CrossRef]

G. L. Cheng, X. M. Hu, W. X. Zhong, and Q. Li, “Two-channel interaction of squeeze-transformed modes with dressed atoms: entanglement enhancement in four-wave mixing in three-level systems,” Phys. Rev. A 78, 033811 (2008).
[CrossRef]

J. Zhang, C. D. Xie, K. C. Peng, and P. van Loock, “Continuous-variable telecloning with phase-conjugate inputs,” Phys. Rev. A 77, 022316 (2008).
[CrossRef]

2007

X. L. Su, A. H. Tan, X. J. Jia, J. Zhang, C. D. Xie, and K. C. Peng, “Experimental preparation of quadripartite cluster and Greenberger–Horne–Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98, 070502 (2007).
[CrossRef]

G. X. Li, H. T. Tan, and M. Macovei, “Enhancement of entanglement for two-mode fields generated from four-wave mixing with the help of the auxiliary atomic transition,” Phys. Rev. A 76, 053827 (2007).
[CrossRef]

S. Pielawa, G. Morigi, D. Vitali, and L. Davidovich, “Generation of Einstein–Podolsky–Rosen entangled radiation through an atomic reservoir,” Phys. Rev. Lett. 98, 240401 (2007).
[CrossRef]

2006

R. Guzmán, J. C. Retamal, E. Solano, and N. Zagury, “Field squeeze operators in optical cavities with atomic ensembles,” Phys. Rev. Lett. 96, 010502 (2006).
[CrossRef]

F. O. Prado, N. G. de Almeida, M. H. Y. Moussa, and C. J. Villas-Bôas, “Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics,” Phys. Rev. A 73, 043803 (2006).
[CrossRef]

M. K. Olsen and A. S. Bradley, “Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions,” Phys. Rev. A 74, 063809 (2006).
[CrossRef]

2005

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

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

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett 94, 100502 (2005).
[CrossRef]

2004

A. Ferraro, M. G. A. Paris, A. Allevi, A. Andreoni, M. Bondani, and E. Puddu, “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241–1249 (2004).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, E. Puddu, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ(2) interactions in noncollinear phase matching,” Opt. Lett. 29, 180–182 (2004).
[CrossRef]

A. Allevi, A. Bondani, A. Ferraro, M. G. A. Paris, and E. Puddu, “Quantum and classical properties of the fields generated by two interlinked second-order non-linear interactions,” J. Mod. Opt. 51, 1031–1036 (2004).
[CrossRef]

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A 69, 063803 (2004).
[CrossRef]

Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004).
[CrossRef]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef]

2003

J. T. Jing, J. Zhang, Y. Yan, F. G. Zhao, C. D. Xie, and K. C. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[CrossRef]

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, “Experimental creation of a fully inseparable tripartite continuous-variable state,” Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[CrossRef]

2001

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

2000

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482–3485 (2000).
[CrossRef]

D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
[CrossRef]

1997

Y. Wu and X. X. Yang, “Effective two-level model for a three-level atom in the cascade configuration,” Phys. Rev. A 56, 2443–2446 (1997).
[CrossRef]

Abdumalikov, A. A.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[CrossRef]

Allevi, A.

Alonso, D.

A. A. Valido, L. A. Correa, and D. Alonso, “Gaussian tripartite entanglement out of equilibrium,” Phys. Rev. A 88, 012309 (2013).
[CrossRef]

Andreoni, A.

Aoki, T.

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, “Experimental creation of a fully inseparable tripartite continuous-variable state,” Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef]

Astafiev, O.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327, 840–843 (2010).
[CrossRef]

Barends, R.

A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmstrøm, J. M. Martinis, and A. N. Cleland, “Planar superconducting resonators with internal quality factors above one million,” Appl. Phys. Lett. 100, 113510 (2012).
[CrossRef]

Baur, M.

M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler–Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
[CrossRef]

Bialczak, R. C.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[CrossRef]

Bianchetti, R.

M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler–Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
[CrossRef]

Blais, A.

J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically induced transparency with amplification in superconducting circuits,” Phys. Rev. Lett. 105, 073601 (2010).
[CrossRef]

M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler–Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
[CrossRef]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef]

Bondani, A.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Resonantly driven Δ-type three-level artificial atom. Two strong fields (ω1,ω2) resonantly drive the atomic transitions |1|2 and |2|3. Three cavity modes (ν1,ν2,ν3), which are far-off resonant with the Rabi sidebands, are generated on the corresponding transitions. (b) The interaction of the dressed atom with the generated cavity modes and (c) the physical model for achieving nonlinear interactions of the simultaneous parametric downconversion process.

Fig. 2.
Fig. 2.

Dynamical evolution of correlation spectra V12, V13, V23 for the cavity modes initially in the vacuum state |0,0,0. The parameters are chosen as Ω1=15g, Ω2=20g, ϕ1=π/2, ϕ2=0, δ=18g, and (a) κ=0.005g, (b) κ=0.002g.

Fig. 3.
Fig. 3.

Dynamical evolution of (a) correlation spectra V12, V13, V23 and (b) the average photon numbers Nl (l=13) of every cavity mode when the cavity modes are initially in the coherent state |10,10,10. The parameters are chosen as Ω1=20g, Ω2=18g, ϕ1=π/2, ϕ2=0, δ=20g, and κ=0.001g.

Fig. 4.
Fig. 4.

Time evolution of correlation spectra V12, V13, V23 for the simultaneous parametric downconversion process when the cavity modes are initially in the vacuum state |0,0,0. The parameters are chosen as (a) Ω1=15g, Ω2=20g, ϕ1=ϕ2=π/6, δ=18g, and κ=0.005g and (b) Ω1=20g, Ω2=18g, ϕ1=ϕ2=π/6, δ=20g, and κ=0.005g.

Equations (13)

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ρ˙=i[ρ,H0+H1]+Lρ,
H0=2[Ω1eiϕ1σ21+Ω2eiϕ2σ32+c.c.],
HI=[g1a1σ31eiδ1t+g2a2σ32eiδ2t+g3a3σ32eiδ3t]+c.c..
Lcρ=l=13κl2(2alρalalalρρalal).
Laρ=D[i<jγijσij]ρ+D[i=2,3γϕiσii]ρ
|+=12(sinθeiϕ2|3+|2+cosθeiϕ1|1),|0=cosθeiϕ1|3sinθeiϕ2|1,|=12(sinθeiϕ2|3|2+cosθeiϕ1|1),
HI=2(g1cos2θei2ϕ1a1+g3cosθeiϕ1a3)σ0+ei(δΩR)t2(g1sin2θei2ϕ2a1+g2cosθeiϕ1a2)σ0ei(δΩR)t+H.c..
Heff=(η1a1a2+η1*a1a2+η2a1a3+η2*a1a3),
ρ˙c=(i/)[ρc,Heff]+Lcρc.
V12=V(X1X2)+V(Y1+Y2+Y3)<4,V13=V(X1X3)+V(Y1+Y2+Y3)<4,V23=V(X2X3)+V(Y1+Y2+Y3)<4.
HI=[g1a1σ31eiδ1t+g2a2σ32eiδ2t+g3a3σ21eiδ3t]+c.c.,
HI=2(g2cosθeiϕ1a2g1sin2θei2ϕ2a1)σ0+ei(δΩR)t+2(g3sinθeiϕ2a3+g1cos2θei2ϕ1a1)σ0ei(δΩR)t+H.c..
Heff=(ζ1a1a2+ζ1*a1a2+ζ2a1a3+ζ2*a1a3),

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