Abstract

Schrödinger cat states, as typical nonclassical states, are very sensitive to the decoherence effects so that swapping these states is a challenge. Here, we propose a reliable scheme to realize the swapping of macroscopic Schrödinger cat state and suppress the decoherence effect in a feedback-controlled optomechanical system that consists of a optical cavity and two mechanical oscillators. Our protocol is composed of three steps. First, we squeeze a mechanical Schrödinger cat state before the state swapping. Then, we complete the state swapping between the two mechanical modes via indirect interaction. Finally, the target mechanical oscillator obtains the Schrödinger cat state by an antisqueezing process. To confirm the superior performance of the protocol, we simulate the whole dynamics of the state transfer and analyze the influence of the squeezed parameters. The corresponding numerical and analytical results show that this approach can be used to reduce the effects of decoherence, which suggests that our state swapping proposal is effective and feasible.

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

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References

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2019 (6)

Y.-X. Zeng, “T. Gebremariam, M.-S. Ding, and C. Li,” Ann. Phys. (Berlin) 531(1), 1970010 (2019).
[Crossref]

F.-Y. Zhang, W.-L. Li, W.-B. Yan, and Y. Xia, “Speeding up adiabatic state conversion in optomechanical systems,” J. Phys. B: At., Mol. Opt. Phys. 52(11), 115501 (2019).
[Crossref]

N. E. Abari, G. V. D. Angelis, S. Zippilli, and D. Vitali, “An optomechanical heat engine with feedback-controlled in-loop light,” New J. Phys. 21(9), 093051 (2019).
[Crossref]

C.-H. Bai, D.-Y. Wang, S. Zhang, S. Liu, and H.-F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, Z. Yang, W.-Z. Zhang, and L. Zhou, “Generation of entangled schrödinger cat state of two macroscopic mirrors,” Opt. Express 27(9), 13547–13558 (2019).
[Crossref]

H. Xie, X. Shang, C.-G. Liao, Z.-H. Chen, and X.-M. Lin, “Macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling,” Phys. Rev. A 100(3), 033803 (2019).
[Crossref]

2018 (7)

J. Li, S. Gröblacher, S.-Y. Zhu, and G. S. Agarwal, “Generation and detection of non-gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
[Crossref]

M. Khazali, “Progress towards macroscopic spin and mechanical superposition via rydberg interaction,” Phys. Rev. A 98(4), 043836 (2018).
[Crossref]

R. A. Brewster, T. B. Pittman, and J. D. Franson, “Reduced decoherence using squeezing, amplification, and antisqueezing,” Phys. Rev. A 98(3), 033818 (2018).
[Crossref]

C. Li, J. Song, Y. Xia, and W. Ding, “Measurement-induced multipartite entanglement for distant four-level atoms in markovian and non-markovian environments,” Phys. Lett. A 382(31), 2044–2048 (2018).
[Crossref]

H. Le Jeannic, A. Cavaillès, K. Huang, R. Filip, and J. Laurat, “Slowing quantum decoherence by squeezing in phase space,” Phys. Rev. Lett. 120(7), 073603 (2018).
[Crossref]

S. Zippilli, N. Kralj, M. Rossi, G. Di Giuseppe, and D. Vitali, “Cavity optomechanics with feedback-controlled in-loop light,” Phys. Rev. A 98(2), 023828 (2018).
[Crossref]

J. D. Franson and R. A. Brewster, “Effects of entanglement in an ideal optical amplifier,” Phys. Lett. A 382(13), 887–893 (2018).
[Crossref]

2017 (5)

K. Johnson, J. Wong-Campos, B. Neyenhuis, J. Mizrahi, and C. Monroe, “Ultrafast creation of large schrödinger cat states of an atom,” Nat. Commun. 8(1), 697 (2017).
[Crossref]

V. Montenegro, R. Coto, V. Eremeev, and M. Orszag, “Macroscopic nonclassical-state preparation via postselection,” Phys. Rev. A 96(5), 053851 (2017).
[Crossref]

W. Li, W. Zhang, C. Li, and H. Song, “Properties and relative measure for quantifying quantum synchronization,” Phys. Rev. E 96(1), 012211 (2017).
[Crossref]

W. Li, C. Li, and H. Song, “Theoretical realization and application of parity-time-symmetric oscillators in a quantum regime,” Phys. Rev. A 95(2), 023827 (2017).
[Crossref]

T. Liu, Y. Zhang, C.-S. Yu, and W.-N. Zhang, “Deterministic transfer of an unknown qutrit state assisted by the low-q microwave resonators,” Phys. Lett. A 381(20), 1727–1731 (2017).
[Crossref]

2016 (5)

J.-Q. Liao, J.-F. Huang, and L. Tian, “Generation of macroscopic schrödinger-cat states in qubit-oscillator systems,” Phys. Rev. A 93(3), 033853 (2016).
[Crossref]

Y. H. Zhou, H. Z. Shen, X. Q. Shao, and X. X. Yi, “Strong photon antibunching with weak second-order nonlinearity under dissipation and coherent driving,” Opt. Express 24(15), 17332–17344 (2016).
[Crossref]

G. D. de Moraes Neto, F. M. Andrade, V. Montenegro, and S. Bose, “Quantum state transfer in optomechanical arrays,” Phys. Rev. A 93(6), 062339 (2016).
[Crossref]

J.-Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref]

W.-W. Zhang, S. K. Goyal, F. Gao, B. C. Sanders, and C. Simon, “Creating cat states in one-dimensional quantum walks using delocalized initial states,” New J. Phys. 18(9), 093025 (2016).
[Crossref]

2015 (1)

E. A. Sete and H. Eleuch, “High-efficiency quantum state transfer and quantum memory using a mechanical oscillator,” Phys. Rev. A 91(3), 032309 (2015).
[Crossref]

2014 (5)

H. Tan, “Deterministic quantum superpositions and fock states of mechanical oscillators via quantum interference in single-photon cavity optomechanics,” Phys. Rev. A 89(5), 053829 (2014).
[Crossref]

C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, “Heralded single-phonon preparation, storage, and readout in cavity optomechanics,” Phys. Rev. Lett. 112(14), 143602 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

S. J. Srinivasan, N. M. Sundaresan, D. Sadri, Y. Liu, J. M. Gambetta, T. Yu, S. M. Girvin, and A. A. Houck, “Time-reversal symmetrization of spontaneous emission for quantum state transfer,” Phys. Rev. A 89(3), 033857 (2014).
[Crossref]

B. T. Kirby and J. D. Franson, “Macroscopic state interferometry over large distances using state discrimination,” Phys. Rev. A 89(3), 033861 (2014).
[Crossref]

2013 (3)

R. Filip, “Gaussian quantum adaptation of non-gaussian states for a lossy channel,” Phys. Rev. A 87(4), 042308 (2013).
[Crossref]

B. T. Kirby and J. D. Franson, “Nonlocal interferometry using macroscopic coherent states and weak nonlinearities,” Phys. Rev. A 87(5), 053822 (2013).
[Crossref]

T. Palomaki, J. Harlow, J. Teufel, R. Simmonds, and K. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature 495(7440), 210–214 (2013).
[Crossref]

2012 (6)

Y.-D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref]

S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484(7393), 195–200 (2012).
[Crossref]

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108(15), 153604 (2012).
[Crossref]

M. Mič, I. Straka, M. Miková, M. Dušek, N. J. Cerf, J. Fiurášek, and M. Ježek, “Noiseless loss suppression in quantum optical communication,” Phys. Rev. Lett. 109(18), 180503 (2012).
[Crossref]

J. Song, Y. Xia, and X.-D. Sun, “Noise-induced quantum correlations via quantum feedback control,” J. Opt. Soc. Am. B 29(3), 268–273 (2012).
[Crossref]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86(6), 063802 (2012).
[Crossref]

2011 (3)

M. Kira, S. Koch, R. Smith, A. Hunter, and S. Cundiff, “Quantum spectroscopy with schrödinger-cat states,” Nat. Phys. 7(10), 799–804 (2011).
[Crossref]

Z.-C. Shi, Y. Xia, J. Song, and H.-S. Song, “Atomic quantum state transferring and swapping via quantum zeno dynamics,” J. Opt. Soc. Am. B 28(12), 2909–2914 (2011).
[Crossref]

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107(22), 220501 (2011).
[Crossref]

2010 (1)

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. Martinis, and A. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464(7289), 697–703 (2010).
[Crossref]

2009 (2)

J. Niset, J. Fiurášek, and N. J. Cerf, “No-go theorem for gaussian quantum error correction,” Phys. Rev. Lett. 102(12), 120501 (2009).
[Crossref]

X. X. Yi, X. L. Huang, C. Wu, and C. H. Oh, “Driving quantum systems into decoherence-free subspaces by lyapunov control,” Phys. Rev. A 80(5), 052316 (2009).
[Crossref]

2007 (1)

K. Jahne, B. Yurke, and U. Gavish, “High-fidelity transfer of an arbitrary quantum state between harmonic oscillators,” Phys. Rev. A 75(1), 010301 (2007).
[Crossref]

2006 (1)

M. Razavi and J. H. Shapiro, “Long-distance quantum communication with neutral atoms,” Phys. Rev. A 73(4), 042303 (2006).
[Crossref]

2004 (2)

D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between matter and light,” Science 306(5696), 663–666 (2004).
[Crossref]

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92(18), 187902 (2004).
[Crossref]

2003 (1)

O. Mandel, M. Greiner, A. Widera, T. Rom, T. Hänsch, and I. Bloch, “Controlled collisions for multi-particle entanglement of optically trapped atoms,” Nature 425(6961), 937–940 (2003).
[Crossref]

2000 (1)

D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley, “Universal fault-tolerant quantum computation on decoherence-free subspaces,” Phys. Rev. Lett. 85(8), 1758–1761 (2000).
[Crossref]

1998 (1)

B. Kane, “A silicon-based nuclear spin quantum computer,” Nature 393(6681), 133–137 (1998).
[Crossref]

1997 (1)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
[Crossref]

1996 (1)

S.-C. Gou, J. Steinbach, and P. L. Knight, “Vibrational pair cat states,” Phys. Rev. A 54(5), 4315–4319 (1996).
[Crossref]

1985 (1)

B. L. Schumaker and C. M. Caves, “New formalism for two-photon quantum optics. ii. mathematical foundation and compact notation,” Phys. Rev. A 31(5), 3093–3111 (1985).
[Crossref]

1963 (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131(6), 2766–2788 (1963).
[Crossref]

Abari, N. E.

N. E. Abari, G. V. D. Angelis, S. Zippilli, and D. Vitali, “An optomechanical heat engine with feedback-controlled in-loop light,” New J. Phys. 21(9), 093051 (2019).
[Crossref]

Abe, H.

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107(22), 220501 (2011).
[Crossref]

Agarwal, G. S.

J. Li, S. Gröblacher, S.-Y. Zhu, and G. S. Agarwal, “Generation and detection of non-gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
[Crossref]

Andrade, F. M.

G. D. de Moraes Neto, F. M. Andrade, V. Montenegro, and S. Bose, “Quantum state transfer in optomechanical arrays,” Phys. Rev. A 93(6), 062339 (2016).
[Crossref]

Angelis, G. V. D.

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[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Sketch of the physical setup to transfer the Schrödinger cat state between optomechanical modes. Two mechanical modes are coupled to an optical cavity pumped by a classical laser field. The quadrature of the optical field is detected via homodyne detection at phase $\theta _ { f b }$ and the corresponding detection results are feedback to the optical cavity.
Fig. 2.
Fig. 2. Sketch of the scheme for QSS of the macroscopic Schrödinger cat state between mechanical modes. In a short duration $\tau$ , the initial mechanical oscillator is squeezed to a squeezed cat state. Then two mechanical oscillators complete state exchange in the time duration $t$ , that is, the target oscillator is prepared in a squeezed cat state. In the last time period $\tau$ , the target mechanical oscillator is antisqueezed to the macroscopic Schrödinger cat state.
Fig. 3.
Fig. 3. (a) Fidelity between the instantaneous state of the system between $\vert 0\rangle _1\vert$ squeezed cat state $\rangle _2$ , i.e., the squeezed cat state exchange between the two mechanical modes. (b) The dynamical evolution of fidelity between the state of the system and the target $\vert 0\rangle _1\vert \textrm {cat}\rangle _2$ . We consider $n^{ef}_1=n^{ef}_2=2$ for the numerical result in (b). The parameters used in panel (a) and (b) are $\gamma _1/\omega ^{\prime \prime }=0.00001$ , $\gamma _2/\omega ^{\prime \prime }=0.00001$ , $\tau =2$ , $\chi =1.5$ , $\mathcal {G}_{11}\beta _1/(2\omega ^{\prime \prime })=\exp (i\pi /2)$ , $\mathcal {G}_{22}\beta _2/(2\omega ^{\prime \prime })=-\exp (i\pi /2)$ , and $\mathcal {G}_{12}\beta _1\beta ^*_2/\omega ^{\prime \prime }=0.05$ .
Fig. 4.
Fig. 4. (a), (b), (c), (d) and (e) are numerical visualization of the Wigner functions of the initial mechanical oscillator at different time, respectively. (f), (g), (h), (i) and (j) represent the Wigner functions of the initial mechanical oscillator at different time, respectively. (a) and (f): $t_1=0$ . (b) and (g): $t_1=2$ . (c) and (h): $t_1=17$ . (d) and (i): $t_1=32$ . (e) and (j): $t_1=34$ . The other parameters are the same with Fig. 3.
Fig. 5.
Fig. 5. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$ under different squeezed strength. The parameters are $\theta =\frac {\pi }{2}$ , $\chi =3$ , $\tau =1$ , $\gamma _{1} / \omega ^{\prime \prime }=0.001$ , $\gamma _{2} / \omega ^{\prime \prime }=0.001$ , and $\mathcal {G}_{12} \beta _{1} \beta _{2}^{*} / \omega ^{\prime \prime }=0.05$ , and $n^{ef}_1=n^{ef}_2=2$ .
Fig. 6.
Fig. 6. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$ under the different squeezed angle. The squeezed strength is $r=1$ . The other parameters are the same as Fig. 5.
Fig. 7.
Fig. 7. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$ , where the squeezed time $\tau =0$ . The other parameters are the same as Fig. 6.
Fig. 8.
Fig. 8. The fidelity of the target state and the instantaneous density matrix as a function of time. The initial state is $\vert \textrm {squeezed cat state}\rangle _1\vert 0\rangle _2$ and the target is $\vert 0\rangle _1\vert \textrm {squeezed cat state}\rangle _2$ . The squeezed angle is $\theta _1=\frac {\pi }{2}$ . The other parameters are $\mathcal {G}_{12} \beta _{1} \beta _{2}^{*} / \omega ^{\prime \prime }=0.05$ , $\chi =4$ , $\gamma _{1} / \omega ^{\prime \prime }=0.005$ , $\gamma _{2} / \omega ^{\prime \prime }=0.005$ , and $n^{ef}_1=n^{ef}_2=2$ .
Fig. 9.
Fig. 9. The fidelity of the target state and the instantaneous density matrix as a function of time. The initial state is $\vert \textrm {squeezed cat state}\rangle _1\vert 0\rangle _2$ and the target is $\vert 0\rangle _1\vert \textrm {squeezed cat state}\rangle _2$ . The squeezed angle is $r_1=1.5$ . The other parameters are the same with Fig. 8.

Equations (47)

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H ^ = H ^ 0 + H ^ I + H ^ d i s s .
H ^ 0 = ω c a ^ a ^ + j = 1 2 ω j b ^ j b ^ j ,
H ^ I = j = 1 , 2 g j a ^ a ^ ( b ^ j + b ^ j ) 2 ,
H ^ d i s s = a ^ ϵ e i Ω t + a ^ ϵ e i Ω t .
H ^ d i s s = Δ a ^ a ^ + j = 1 2 ω j b ^ j b ^ j + g j a ^ a ^ ( b ^ j + b ^ j ) 2 + a ^ ϵ + a ^ ϵ .
H ^ d i s s = Δ a ^ a ^ + j = 1 2 ω j b ^ j b ^ j + 2 g j a ^ a ^ b ^ j b ^ j + a ^ ϵ + a ^ ϵ ,
H ^ d i s s = Δ a ^ a ^ + j = 1 2 ω j b ^ j b ^ j + 2 g j { ( α β j a ^ b ^ j + α β j a ^ b ^ j ) + H . c } ,
a ^ ˙ = ( κ i Δ ) a ^ i j = 1 2 G j α ( β j b ^ j + β j b ^ j ) + 2 κ a ^ i n , b ^ ˙ j = ( γ j + i ω j ) b ^ j i G j β j ( α a ^ + α a ^ ) + 2 γ j b ^ i n , j ,
a ^ i n = 2 κ 1 a ^ i n , 1 + 2 κ 2 a ^ i n , 2 2 κ .
a ^ i n ( 1 ) ( t ) = a ^ i n , 0 ( t ) + Φ ^ 1 ( t ) ,
i ^ f b ( t ) = η d X ^ o u t , f b ( θ f b ) ( t ) + 1 η d X ^ v ( t ) ,
a ^ o u t , 2 ( t ) = 2 κ 2 a ^ ( t ) a ^ i n , 2 ( t ) .
a ^ o u t , 2 ( t τ f b ) a ¯ ^ o u t , 2 ( t ) e i Δ t e i Δ τ f b = a ^ o u t , 2 ( t ) e i Δ τ f b ,
a ^ ˙ ( t ) = ( κ f b i Δ ) a ^ ( t ) i j G j α ( β b ^ j ( t ) + β b ^ j ( t ) ) + 2 ξ a ^ ( t ) + 2 κ f b a ^ i n , f b ( t ) ,
a ^ i n , f b ( t ) = 1 2 κ f b { 2 κ 1 a ^ i n , 0 ( t ) + 2 κ 2 a ^ i n , 2 ( t ) g ¯ f b 2 η d κ 1 [ a ^ i n , 2 ( t ) + a ^ i n , 2 ( t ) ] + g ¯ f b 2 ( 1 η d ) κ 1 X ^ v ( t ) } .
a ^ i n , f b ( t ) a ^ i n , f b ( t ) = n o p t , f b δ ( t t ) ,
n o p t , f b = ( κ κ f b ) 2 4 η d κ 2 κ f b .
H ^ e f f = Δ a ^ a ^ + i ( ξ a ^ 2 ξ a ^ 2 ) + i = 1 2 ω j b ^ m , i b ^ m , i + G i ( β b ^ i + β b ^ i ) ( α a ^ i + α a ^ i ) .
H ^ e f f = j = 1 , 2 G j j ¯ 2 ( β j b ^ j + β j b ^ j ) ( β j ¯ b ^ j ¯ + β j ¯ b ^ j ¯ ) + ω j b ^ j b ^ j + G j j 2 ( β j b ^ j 2 + β j b ^ j 2 ) ,
H ^ e 1 = ω 2 b ^ 2 b ^ 2 + j = 1 , 2 i γ j b ^ j b ^ j + G 11 2 ( β 1 b ^ 1 2 + β 1 b ^ 1 2 ) .
H ^ e 2 = j = 1 , 2 G j j ¯ 2 ( β j β j ¯ b ^ j b ^ j ¯ + β j β j ¯ b ^ j b ^ j ¯ ) + ( ω j i γ j ) b ^ j b ^ j ,   ( j = 1 , 2 ,   j ¯ = 2 , 1 ) ,
H ^ e 3 = ω 1 b ^ 1 b ^ 1 + j = 1 , 2 i γ j b ^ j b ^ j + G 22 2 ( β 2 b ^ 2 2 + β 2 b ^ 2 2 ) .
| c a t = 1 2 ( | χ + e i ϑ | χ ) ,
F = | ϕ t | e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | ϕ i | = 1 4 | χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 + e i θ χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 + e i θ χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 + χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 | .
| squeezed cat state 1 | 0 2 = S 1 ( r 1 e i θ 1 ) | cat 1 | 0 1 , | 0 1 | squeezed cat state 2 = S 2 ( r 1 e i θ 1 ) | 0 1 | cat 2 ,
S ( x ) = exp { i ( x b ^ j 2 + x b ^ j 2 ) } ,     j = 1 , 2.
0 = ( κ f b i Δ ) a ^ j i G j α ( β j b ^ j + β j b ^ j ) + 2 ξ a ^ + 2 κ b f a ^ i n , b f .
a ^ = j i B j ( β j b ^ j + β j b ^ j ) + 2 κ f b ( U a ^ i n , f b + V a ^ i n , f b ) ,
B j = G j ( U α V α ) ,                       U = κ κ b f Δ κ 2 + 2 κ κ b f ,                       V = κ f b + i Δ Δ κ 2 + 2 κ κ b f .
b ^ ˙ j = ( γ j + i ω j ) b ^ j i G j j ¯ β j ( β j ¯ b ^ j ¯ + β j ¯ b ^ j ¯ ) i G j j β j b ^ j + 2 γ j b ^ i n , j e f ,
ω j = ω j + G j j | β j | 2 ,                   G j k = 2 G j G k ( ( μ ) | α | 2 + ν ( α 2 ) ) ,                   μ = κ + i Δ Δ 2 + 2 κ b f κ κ b f 2 , ν = κ κ f b Δ 2 + 2 κ b f κ κ b f 2 ,             b ^ j , i n = l a ^ i n , f b + H . c . ,             b ^ j , i n e f f = b ^ i n , j i G j β j κ f b γ j b ^ j , i n , l = ν α + μ α .
H ^ e f f = j = 1 , 2 ω j b ^ j b ^ j + G j j ¯ 2 ( β j b ^ j + β j b ^ j ) ( β j ¯ b ^ j ¯ + β j ¯ b ^ j ¯ ) + G j j 2 ( β j b ^ j 2 + β j b ^ j 2 ) .
ρ ^ ˙ = i [ H ^ f b ( t ) , ρ ^ ] + j = 1 , 2 D [ b ^ j ] ,
D [ b ^ j ] = γ j ( n t h , j e f + 1 ) ( 2 b ^ j ρ ^ b ^ j b ^ j b ^ j ρ ^ ρ ^ b ^ j b ^ j ) + γ j n t h , j e f ( 2 b ^ j ρ ^ b ^ j b ^ j b ^ j ρ ^ ρ ^ b ^ j b ^ j ) ,
n t h , j e f = n t h , j + G j 2 | β j | 2 κ f b γ j { | l | 2 ( 2 n o p t , f b + 1 ) + 2 ( l 2 ) n o p t , f b ( 1 η d κ 2 κ 1 ) } .
e i H ^ 1 τ = e ζ 0 ( 1 ) e ζ 1 ( 1 ) b ^ 2 b ^ 2 e ζ + ( 1 ) b ^ 1 2 e ln ( ζ 3 ( 1 ) ) 2 b ^ 1 b ^ 1 e ζ ( 1 ) b ^ 1 2 ,
η + ( 1 ) = i G 11 β 1 τ ,                       η ( 1 ) = i G 11 β 1 τ ,                       η 3 ( 1 ) = 2 γ 1 τ ,
ζ 0 ( 1 ) = 2 γ 1 τ + ln ( ζ 3 ( 1 ) ) 4 ,                                         ζ 1 ( 1 ) = ( i ω 2 γ 2 ) τ ,                               ζ 3 ( 1 ) = ( cosh ϵ 1 η 3 ( 1 ) 2 ϵ 1 sinh ϵ 1 ) 2 , ζ ± ( 1 ) = η ± ( 1 ) sinh ϵ 1 2 ϵ 1 cosh ϵ 1 η 3 ( 1 ) sinh ϵ 1 ,                   ϵ 1 2 = 1 4 ( η 3 ( 1 ) ) 2 η + ( 1 ) η ( 1 ) .
e i H ^ 2 t = e ζ 0 ( 2 ) ( b ^ 1 b ^ 1 + b ^ 2 b ^ 2 ) e ζ + ( 2 ) b ^ 1 b ^ 2 e ln ( ζ 3 ( 2 ) ) 2 ( b ^ 1 b ^ 1 b ^ 2 b ^ 2 ) e ζ ( 2 ) b ^ 1 b ^ 2 ,
η + ( 2 ) = i G 12 β 1 β 2 t ,                     η ( 2 ) = i G 12 β 1 β 2 t ,                       ζ 0 ( 2 ) = ( i ω 1 + γ 1 ) t , ζ 3 ( 2 ) = ( cosh ϵ 2 ) 2 ,                       ζ ± ( 2 ) = η ± ( 2 ) sinh ϵ 2 ϵ 2 cosh ϵ 2 ,                       ϵ 2 2 = η + ( 2 ) η ( 2 ) .
e i H ^ 3 τ = e ζ 0 ( 3 ) e ζ 1 ( 3 ) b ^ 1 b ^ 1 e ζ + ( 3 ) b ^ 2 2 e ln ( ζ 3 ( 3 ) ) 2 b ^ 2 b ^ 2 e ζ ( 3 ) b ^ 2 2 ,
η + ( 3 ) = i G 22 β 2 τ ,                       η ( 3 ) = i G 22 β 2 τ ,                       η 3 ( 3 ) = 2 γ 2 τ ,
ζ 0 ( 3 ) = 2 γ 2 τ + ln ( ζ 3 ( 3 ) ) 4 ,                                         ζ 1 ( 3 ) = ( i ω 1 γ 1 ) τ ,                       ζ 3 ( 3 ) = ( cosh ϵ 3 η 3 ( 3 ) 2 ϵ 3 sinh ϵ 3 ) 2 , ζ ± ( 3 ) = η ± ( 3 ) sinh ϵ 3 2 ϵ 3 cosh ϵ 3 η 3 ( 3 ) sinh ϵ 3 ,                       ϵ 3 2 = 1 4 ( η 3 ( 3 ) ) 2 η + ( 3 ) η ( 3 ) .
χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 = e ( ζ 0 ( 1 ) + ζ 0 ( 3 ) + ζ ( 1 ) χ 2 + | χ | 2 2 ( | ζ 3 ( 1 ) | 2 1 ) + ζ + ( 3 ) χ 2 ) e | χ | 2 2 ( | ζ 3 ( 3 ) | 2 1 ) ) × e ( χ ζ ( 2 ) ζ 3 ( 3 ) ζ 3 ( 2 ) e ζ 0 ( 2 ) χ ζ 3 ( 1 ) ) e ζ + ( 1 ) ( ζ ( 2 ) ) 2 χ 2 ζ 3 ( 1 ) ζ 3 ( 3 ) e 2 ζ 0 ( 2 ) ,
χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 = e ( ζ 0 ( 1 ) + ζ 0 ( 3 ) + ζ ( 1 ) χ 2 + | χ | 2 2 ( | ζ 3 ( 1 ) | 2 1 ) + ζ + ( 3 ) χ 2 ) e | χ | 2 2 ( | ζ 3 ( 3 ) | 2 1 ) ) × e ( χ ζ ( 2 ) ζ 3 ( 3 ) ζ 3 ( 2 ) e ζ 0 ( 2 ) χ ζ 3 ( 1 ) ) e ζ + ( 1 ) ( ζ ( 2 ) ) 2 χ 2 ζ 3 ( 1 ) ζ 3 ( 3 ) e 2 ζ 0 ( 2 ) ,
χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 = e ( ζ 0 ( 1 ) + ζ 0 ( 3 ) + ζ ( 1 ) χ 2 + | χ | 2 2 ( | ζ 3 ( 1 ) | 2 1 ) + ζ + ( 3 ) χ 2 ) e | χ | 2 2 ( | ζ 3 ( 3 ) | 2 1 ) ) × e ( χ ζ ( 2 ) ζ 3 ( 3 ) ζ 3 ( 2 ) e ζ 0 ( 2 ) χ ζ 3 ( 1 ) ) e ζ + ( 1 ) ( ζ ( 2 ) ) 2 χ 2 ζ 3 ( 1 ) ζ 3 ( 3 ) e 2 ζ 0 ( 2 ) ,
χ | 2 0 | 1 e i H ^ 3 τ e i H ^ 2 t e i H ^ 1 τ | χ 1 | 0 2 = e ( ζ 0 ( 1 ) + ζ 0 ( 3 ) + ζ ( 1 ) χ 2 + | χ | 2 2 ( | ζ 3 ( 1 ) | 2 1 ) + ζ + ( 3 ) χ 2 ) e | χ | 2 2 ( | ζ 3 ( 3 ) | 2 1 ) ) × e ( χ ζ ( 2 ) ζ 3 ( 3 ) ζ 3 ( 2 ) e ζ 0 ( 2 ) χ ζ 3 ( 1 ) ) e ζ + ( 1 ) ( ζ ( 2 ) ) 2 χ 2 ζ 3 ( 1 ) ζ 3 ( 3 ) e 2 ζ 0 ( 2 ) .

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