Abstract

We present two efficient methods for implementing the Fredkin gate with atoms separately trapped in an array of three high-Q coupled cavities. The first proposal is based on the resonant dynamics, which leads to a fast resonant interaction in a certain subspace while leaving others unchanged, and the second one utilizes a dispersive interaction such that an effective long-distance dipole–dipole interaction between two distributed target qubits is achieved by a virtually excited process. Both schemes can achieve the standard form of the Fredkin gate in a single step without any subsequent single-qubit operation. The effects of decoherence on the performance of the gate are also analyzed in virtue of the master equation, and strictly numerical simulation reveals that the average fidelity of the quantum gate is high.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  46. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
    [CrossRef]
  47. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
    [CrossRef]
  48. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007).
    [CrossRef]
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2013

C. Jones, “Composite Toffoli gate with two-round error detection,” Phys. Rev. A 87, 052334 (2013).
[CrossRef]

S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
[CrossRef]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[CrossRef]

2012

X. Q. Shao, T. Y. Zheng, and S. Zhang, “Fast synthesis of the Fredkin gate via quantum Zeno dynamics,” Quant. Info. Proc. 11, 1797–1808 (2012).

V. M. Stojanović, A. Fedorov, A. Wallraff, and C. Bruder, “Quantum-control approach to realizing a Toffoli gate in circuit QED,” Phys. Rev. B 85, 054504 (2012).

A. M. Chen, S. Y. Cho, and M. D. Kim, “Implementation of a three-qubit Toffoli gate in a single step,” Phys. Rev. A 85, 032326 (2012).
[CrossRef]

X. Q. Shao, T. Y. Zheng, and S. Zhang, “Robust Toffoli gate originating from Stark shifts,” J. Opt. Soc. Am. B 29, 1203–1207 (2012).
[CrossRef]

2010

W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
[CrossRef]

C. P. Yang, Y. X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

C. P. Yang, S. B. Zheng, and F. Nori, “Multiqubit tunable phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 82, 062326 (2010).
[CrossRef]

Z. B. Yang, Y. Xia, and S. B. Zheng, “Resonant scheme for realizing quantum phase gates for two separate atoms via coupled cavities,” Opt. Commun. 283, 3052–3057 (2010).
[CrossRef]

2009

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the quantum Toffoli gate with trapped ions,” Phys. Rev. Lett. 102, 040501 (2009).
[CrossRef]

G. W. Lin, X. B. Zou, X. M. Lin, and G. C. Guo, “Robust and fast geometric quantum computation with multiqubit gates in cavity QED,” Phys. Rev. A 79, 064303 (2009).
[CrossRef]

X. Q. Shao, H. F. Wang, L. Chen, S. Zhang, and K. H. Yeon, “One-step implementation of the Toffoli gate via quantum Zeno dynamics,” Phys. Lett. A 374, 28–33 (2009).
[CrossRef]

Z. B. Yang, H. Z. Wu, W. J. Su, and S. B. Zheng, “Quantum phase gates for two atoms trapped in separate cavities within the null- and single-excitation subspaces,” Phys. Rev. A 80, 012305 (2009).
[CrossRef]

Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79, 022301 (2009).
[CrossRef]

Q. Lin and B. He, “Single-photon logic gates using minimal resources,” Phys. Rev. A 80, 042310 (2009).
[CrossRef]

J. Song, Y. Xia, and H. S. Song, “Quantum gate operations using atomic qubits through cavity input-output process,” Europhys. Lett. 87, 50005 (2009).
[CrossRef]

2008

J. Fiurášek, “Linear optical Fredkin gate based on partial-SWAP gate,” Phys. Rev. A 78, 032317 (2008).
[CrossRef]

Y. X. Gong, G. C. Guo, and T. C. Ralph, “Methods for a linear optical quantum Fredkin gate,” Phys. Rev. A 78, 012305 (2008).
[CrossRef]

M. J. Hartmann, F. G. S. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photonics Rev. 2, 527–556 (2008), and reference therein.
[CrossRef]

E. K. Irish, C. D. Ogden, and M. S. Kim, “Polaritonic characteristics of insulator and superfluid states in a coupled-cavity array,” Phys. Rev. A 77, 033801 (2008).
[CrossRef]

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2, 741–747 (2008).
[CrossRef]

2007

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007).
[CrossRef]

A. G. White, A. Gilchrist, G. J. Pryde, J. L. O’Brien, M. J. Bremner, and N. K. Langford, “Measuring two-qubit gates,” J. Opt. Soc. Am. B 24, 172–183 (2007).
[CrossRef]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[CrossRef]

Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007).
[CrossRef]

B. Wang and L. M. Duan, “Implementation scheme of controlled SWAP gates for quantum fingerprinting and photonic quantum computation,” Phys. Rev. A 75, 050304(R) (2007).
[CrossRef]

2006

C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
[CrossRef]

J. Fiurášek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
[CrossRef]

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[CrossRef]

2005

C. P. Yang and S. Han, “n-qubit-controlled phase gate with superconducting quantum-interference devices coupled to a resonator,” Phys. Rev. A 72, 032311 (2005).
[CrossRef]

A. Beige, H. Cable, C. Marr, and P. L. Knight, “Speeding up gate operations through dissipation,” Laser Phys. 15, 162–169 (2005).

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

2003

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

2002

M. A. Nielsen, “A simple formula for the average gate fidelity of a quantum dynamical operation,” Phys. Lett. A 303, 249–252 (2002).
[CrossRef]

P. Facchi and S. Pascazio, “Quantum Zeno dynamics of a field in a cavity,” Phys. Rev. Lett. 89, 080401 (2002).
[CrossRef]

2001

X. B. Zou, J. Kim, and H. W. Lee, “Generation of two-mode nonclassical motional states and a Fredkin gate operation in a two-dimensional ion trap,” Phys. Rev. A 63, 065801 (2001).
[CrossRef]

1998

L. K. Grover, “Quantum computers can search rapidly by using almost any transformation,” Phys. Rev. Lett. 80, 4329–4332 (1998).
[CrossRef]

1997

P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26, 1484–1509 (1997).
[CrossRef]

A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, “Stabilisation of quantum computations by symmetrisation,” SIAM J. Comput. 26, 1541–1557 (1997).
[CrossRef]

1995

I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995).
[CrossRef]

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[CrossRef]

1989

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62, 2124–2127 (1989).
[CrossRef]

1982

E. Fredkin and T. Toffoli, “Conservative logic,” Int. J. Theor. Phys. 21, 219–253 (1982).
[CrossRef]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Angelakis, D. G.

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[CrossRef]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Barenco, A.

A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, “Stabilisation of quantum computations by symmetrisation,” SIAM J. Comput. 26, 1541–1557 (1997).
[CrossRef]

Beige, A.

A. Beige, H. Cable, C. Marr, and P. L. Knight, “Speeding up gate operations through dissipation,” Laser Phys. 15, 162–169 (2005).

Berthiaume, A.

A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, “Stabilisation of quantum computations by symmetrisation,” SIAM J. Comput. 26, 1541–1557 (1997).
[CrossRef]

Blatt, R.

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the quantum Toffoli gate with trapped ions,” Phys. Rev. Lett. 102, 040501 (2009).
[CrossRef]

Bose, S.

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[CrossRef]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007).
[CrossRef]

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[CrossRef]

Brandão, F. G. S.

M. J. Hartmann, F. G. S. Brandão, and M. B. Plenio, “Quantum many-body phenomena in coupled cavity arrays,” Laser Photonics Rev. 2, 527–556 (2008), and reference therein.
[CrossRef]

Bremner, M. J.

Bruder, C.

V. M. Stojanović, A. Fedorov, A. Wallraff, and C. Bruder, “Quantum-control approach to realizing a Toffoli gate in circuit QED,” Phys. Rev. B 85, 054504 (2012).

Cable, H.

A. Beige, H. Cable, C. Marr, and P. L. Knight, “Speeding up gate operations through dissipation,” Laser Phys. 15, 162–169 (2005).

Chen, A. M.

A. M. Chen, S. Y. Cho, and M. D. Kim, “Implementation of a three-qubit Toffoli gate in a single step,” Phys. Rev. A 85, 032326 (2012).
[CrossRef]

Chen, L.

X. Q. Shao, H. F. Wang, L. Chen, S. Zhang, and K. H. Yeon, “One-step implementation of the Toffoli gate via quantum Zeno dynamics,” Phys. Lett. A 374, 28–33 (2009).
[CrossRef]

Cho, J.

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[CrossRef]

Cho, S. Y.

A. M. Chen, S. Y. Cho, and M. D. Kim, “Implementation of a three-qubit Toffoli gate in a single step,” Phys. Rev. A 85, 032326 (2012).
[CrossRef]

Chuang, I. L.

I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995).
[CrossRef]

Chwalla, M.

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the quantum Toffoli gate with trapped ions,” Phys. Rev. Lett. 102, 040501 (2009).
[CrossRef]

Deutsch, D.

A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, “Stabilisation of quantum computations by symmetrisation,” SIAM J. Comput. 26, 1541–1557 (1997).
[CrossRef]

DiVincenzo, D. P.

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[CrossRef]

Du, J. F.

W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
[CrossRef]

Duan, L. M.

B. Wang and L. M. Duan, “Implementation scheme of controlled SWAP gates for quantum fingerprinting and photonic quantum computation,” Phys. Rev. A 75, 050304(R) (2007).
[CrossRef]

Ekert, A.

A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, “Stabilisation of quantum computations by symmetrisation,” SIAM J. Comput. 26, 1541–1557 (1997).
[CrossRef]

Facchi, P.

P. Facchi and S. Pascazio, “Quantum Zeno dynamics of a field in a cavity,” Phys. Rev. Lett. 89, 080401 (2002).
[CrossRef]

Fedorov, A.

V. M. Stojanović, A. Fedorov, A. Wallraff, and C. Bruder, “Quantum-control approach to realizing a Toffoli gate in circuit QED,” Phys. Rev. B 85, 054504 (2012).

Feng, M.

W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
[CrossRef]

Fiurášek, J.

J. Fiurášek, “Linear optical Fredkin gate based on partial-SWAP gate,” Phys. Rev. A 78, 032317 (2008).
[CrossRef]

J. Fiurášek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
[CrossRef]

Fredkin, E.

E. Fredkin and T. Toffoli, “Conservative logic,” Int. J. Theor. Phys. 21, 219–253 (1982).
[CrossRef]

Gilchrist, A.

Goh, K. W.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Gong, Y. X.

Y. X. Gong, G. C. Guo, and T. C. Ralph, “Methods for a linear optical quantum Fredkin gate,” Phys. Rev. A 78, 012305 (2008).
[CrossRef]

Grover, L. K.

L. K. Grover, “Quantum computers can search rapidly by using almost any transformation,” Phys. Rev. Lett. 80, 4329–4332 (1998).
[CrossRef]

Guo, G. C.

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W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
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X. Q. Shao, T. Y. Zheng, and S. Zhang, “Fast synthesis of the Fredkin gate via quantum Zeno dynamics,” Quant. Info. Proc. 11, 1797–1808 (2012).

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J. Song, Y. Xia, and H. S. Song, “Quantum gate operations using atomic qubits through cavity input-output process,” Europhys. Lett. 87, 50005 (2009).
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X. Q. Shao, H. F. Wang, L. Chen, S. Zhang, and K. H. Yeon, “One-step implementation of the Toffoli gate via quantum Zeno dynamics,” Phys. Lett. A 374, 28–33 (2009).
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G. W. Lin, X. B. Zou, X. M. Lin, and G. C. Guo, “Robust and fast geometric quantum computation with multiqubit gates in cavity QED,” Phys. Rev. A 79, 064303 (2009).
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Figures (5)

Fig. 1.
Fig. 1.

Schematic of three atoms trapped in three coupled cavities. Each qubit is encoded into two lower-energy levels labeled as |0 and |1. The transitions between the levels |ei|0i are coupled to the cavity mode with the coupling constants gi, and the transitions |e1(3)|11(3) are driven by a classical pulse with the Rabi frequencies Ω1(3). Δ represents the corresponding one-photon detuning parameter, and the photon can hop between two cavities with coupling strength J. The atom in the middle cavity serves as the control qubit, while the atoms in bilateral cavities play the role of the target qubits. We label the atoms as 1, 2, 3 from left to right for convenience, and the state |α1,β2,γ3a is abbreviated to |α,β,γa in the text.

Fig. 2.
Fig. 2.

Evolutions for the populations of interested states. The qubit states |011a|000c and |110a|000c coherently transform into each other via the media state |D, while other qubit states are frozen to their original status due to the large detuned interaction. The corresponding parameters are J=g and Ωmax=0.05g.

Fig. 3.
Fig. 3.

Evolution of the system from exact calculations of Hamiltonian Eq. (3). (a) These two curves represent the conversion of populations between states |001 and |100. (b) Populations of states |011 and |110. (c) Population of state |101. (d) Population of state |111. All cavities stay in vacuum states, and the data listed in the data cursor further validate the effectiveness of our approximation. The revelant parameters are set as Ω1=Ω3=Ω=0.02g and Δ=J=g. The dimensionless time is measured in units of g1.

Fig. 4.
Fig. 4.

Average gate fidelity versus the Zeno requirement Ωmax(Ω)/g. The black curve represents the resonant model, and the red curve corresponds to the dispersive model.

Fig. 5.
Fig. 5.

Average gate fidelities for resonant model and dispersive model versus decoherence parameter κ/g under three different driving lasers, respectively, where we have set γ=κ and Δ=J=g.

Equations (19)

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UFRED=[1000000001000000001000000001000000001000000000100000010000000001].
HS=j=1,3Ωj(|ej1j|eiωljt+|1jej|eiωljt)+k=12J(akak+1+akak+1)+i=13gi(ai|ei0i|+|0iei|ai)+k=13ωcakak+i=13ωe|eiei|+ω0|0i0i|+ω1|1i1i|,
HI=j=1,3Ωj(|ej1j|+|1jej|)+k=12J(akak+1+akak+1)+i=13gi(ai|ei0i|+|0iei|ai)+Δ|eiei|.
HI=H1+H2,H1=j=1,3Ωj(|ej1j|+|1jej|),H2=k=12J(akak+1+akak+1)+i=13gi(ai|ei0i|+|0iei|ai),
Heffre=nηnPn+PnH1Pn.
Heffre=P0H1P0.
Mre=[0Ω300000Ω30g00000g0J00000J0J00000J0g00000g0Ω100000Ω10].
Hre=Ω1|110a|000c[13E1|+12(E2|+E3|)112(E4|E5|)]+Ω3|011a|000c[13E1|12(E2|+E3|)112(E4|E5|)]+H.c.+i=15Ei|EiEi|.
Heffre=13(Ω1|110a+Ω3|011a)|000cD|+H.c.,
0T13Ωdt=π2,(Ω1=Ω3=Ω).
Ω(t)=2Ωmaxsin2[23Ωmaxt],
Mde=[Δg0000g2J000000Δg0000g2J000000Δg0000g0],
E1=12(Δ2J4g2+Δ2+22ΔJ+2J2),E2=12(Δ2J+4g2+Δ2+22ΔJ+2J2),E3=12(Δ+2J4g2+Δ222ΔJ+2J2),E4=12(Δ+2J+4g2+Δ222ΔJ+2J2),E5=12(Δ4g2+Δ2),
|E1=α4g2+α2|φ1+2g4g2+α2|φa,|E2=2g4g2+α2|φ1α4g2+α2|φa,|E3=β4g2+β2|φ2+2g4g2+β2|φb,|E4=2g4g2+β2|φ2β4g2+β2|φb,|E5=4g2+Δ2Δ24g2+Δ2|φ3+4g2+Δ2+Δ24g2+Δ2|φc,|E6=4g2+Δ2+Δ24g2+Δ2|φ3+4g2+Δ2Δ24g2+Δ2|φc,
α=Δ+2J4g2+Δ2+22ΔJ+2J2,β=Δ2J4g2+Δ222ΔJ+2J2.
Heff1=Ω12g|100100|+Ω32g|001001|+Ω1Ω3g(|100001|+H.c.).
Heff2=Ω122g|110110|Ω322g|011011|Ω1Ω32g(|110011|+H.c.).
F¯(ε,UFRED)=jtr[UFREDUjUFREDε(Uj)]+d2d2(d+1),
ρ˙=i[HI,ρ]i=13κ2(aiaiρ2aiρai+ρaiai)j=0,1n=13γnej2(σeenρ2σjenρσejn+ρσeen),

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