Abstract

In this paper, we provide an analytical investigation of the entanglement dynamics of moving qubits dissipating into a common and (in general) non-Markovian environment for both weak and strong coupling regimes. We first consider the case of two moving qubits in a common environment and then generalize it to an arbitrary number of moving qubits. Our results show that when the system evolves from an initial entangled state, the amount of initial entanglement decreases and finally disappears after a finite interval of time due to the environmental effects. Moreover, we observe that the movement of qubits has a constructive role in the protection of the initial entanglement. In a sense, in this case, we observe a Zeno-like effect due to the velocity of qubits. On the other hand, we demonstrate how a stationary state of entanglement may be achieved when we consider the case in which at least one of the moving qubits is initially in the ground state. Surprisingly, we observe that when we extend the number of moving qubits with the same velocity, the stationary state of the qubits does not depend on the velocity of qubits as well as on the environmental properties. This means that, in this condition, the stationary state of entanglement depends only on the number of moving qubits.

© 2020 Optical Society of America

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2019 (3)

G. Sorelli, N. Leonhard, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, “Entanglement protection of high-dimensional states by adaptive optics,” New J. Phys. 21, 023003 (2019).
[Crossref]

D. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Müller, P. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

S. Golkar and M. K. Tavassoly, “Atomic motion and dipole-dipole effects on the stability of atom-atom entanglement in Markovian/non-Markovian reservoir,” Mod. Phys. Lett. A 34, 1950077 (2019).
[Crossref]

2018 (8)

N. Behzadi, B. Ahansaz, E. Faizi, and H. Kasani, “Requirement of system-reservoir bound states for entanglement protection,” Quantum Inf. Process. 17, 65 (2018).
[Crossref]

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. Devoret, L. Jiang, and R. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018).
[Crossref]

D. Layden and P. Cappellaro, “Spatial noise filtering through error correction for quantum sensing,” NPJ Quantum Inf. 4, 30 (2018).
[Crossref]

A. Mortezapour, G. Naeimi, and R. L. Franco, “Coherence and entanglement dynamics of vibrating qubits,” Opt. Commun. 424, 26–31 (2018).
[Crossref]

S. Golkar and M. K. Tavassoly, “Coping with attenuation of quantum correlations of two qubit systems in dissipative environments: multi-photon transitions,” Eur. Phys. J. D 72, 184 (2018).
[Crossref]

S. Golkar and M. K. Tavassoly, “Dynamics and maintenance of bipartite entanglement via the stark shift effect inside dissipative reservoirs,” Laser Phys. Lett. 15, 035205 (2018).
[Crossref]

A. Mortezapour and R. L. Franco, “Protecting quantum resources via frequency modulation of qubits in leaky cavities,” Sci. Rep. 8, 14304 (2018).
[Crossref]

S. Golkar and M. K. Tavassoly, “Dynamics of entanglement protection of two qubits using a driven laser field and detunings: independent and common, Markovian and/or non-Markovian regimes,” Chin. Phys. B 27, 040303 (2018).
[Crossref]

2017 (10)

M. Rafiee, A. Nourmandipour, and S. Mancini, “Universal feedback control of two-qubit entanglement,” Phys. Rev. A 96, 012340 (2017).
[Crossref]

M. Ghasemi, M. K. Tavassoly, and A. Nourmandipour, “Dissipative entanglement swapping in the presence of detuning and Kerr medium: Bell state measurement method,” Eur. Phys. J. Plus 132, 531 (2017).
[Crossref]

G. Calajó and P. Rabl, “Strong coupling between moving atoms and slow-light Cherenkov photons,” Phys. Rev. A 95, 043824 (2017).
[Crossref]

D. Moustos and C. Anastopoulos, “Non-Markovian time evolution of an accelerated qubit,” Phys. Rev. D 95, 025020 (2017).
[Crossref]

A. Mortezapour, M. A. Borji, and R. L. Franco, “Protecting entanglement by adjusting the velocities of moving qubits inside non-Markovian environments,” Laser Phys. Lett. 14, 055201 (2017).
[Crossref]

N. Behzadi, E. Faizi, and O. Heibati, “Quantum discord protection of a two-qutrit V-type atomic system from decoherence by partially collapsing measurements,” Quantum Inf. Process. 16, 257 (2017).
[Crossref]

A. Mortezapour, M. A. Borji, D. Park, and R. L. Franco, “Non-Markovianity and coherence of a moving qubit inside a leaky cavity,” Open Syst. Inf. Dyn. 24, 1740006 (2017).
[Crossref]

S. Mirza-Zadeh, M. Saadati-Niari, and M. Amniat-Talab, “Creation of n-partite W-states by adiabatic passage and pulse area techniques,” J. Mod. Opt. 64, 2376–2384 (2017).
[Crossref]

N. Behzadi, B. Ahansaz, and E. Faizi, “Quantum coherence and entanglement preservation in Markovian and non-Markovian dynamics via additional qubits,” Eur. Phys. J. D 71, 280 (2017).
[Crossref]

L. García-Álvarez, S. Felicetti, E. Rico, E. Solano, and C. Sabín, “Entanglement of superconducting qubits via acceleration radiation,” Sci. Rep. 7, 657 (2017).
[Crossref]

2016 (4)

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94, 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93, 022327 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. A. Bolorizadeh, “Quantum Zeno and anti-Zeno effects on the entanglement dynamics of qubits dissipating into a common and non-Markovian environment,” J. Opt. Soc. Am. B 33, 1723–1730 (2016).
[Crossref]

M. Rafiee, A. Nourmandipour, and S. Mancini, “Optimal feedback control of two-qubit entanglement in dissipative environments,” Phys. Rev. A 94, 012310 (2016).
[Crossref]

2015 (5)

A. Nourmandipour and M. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B 48, 165502 (2015).
[Crossref]

L. Aolita, F. de Melo, and L. Davidovich, “Open-system dynamics of entanglement: a key issues review,” Rep. Prog. Phys. 78, 042001 (2015).
[Crossref]

M. Flores and E. Galapon, “Two qubit entanglement preservation through the addition of qubits,” Ann. Phys. 354, 21–30 (2015).
[Crossref]

H. R. Jahromi and M. Amniat-Talab, “Noncyclic geometric quantum computation and preservation of entanglement for a two-qubit Ising model,” Quantum Inf. Process. 14, 3739–3755 (2015).
[Crossref]

X.-L. Zong, C.-Q. Du, M. Yang, W. Song, Q. Yang, and Z.-L. Cao, “Protecting multipartite entanglement against weak-measurement-induced amplitude damping by local unitary operations,” Quantum Inf. Process. 14, 3423–3440 (2015).
[Crossref]

2014 (2)

R. Ghanbari and M. Rafiee, “Stationary entanglement and discord for dissipating qubits by local magnetic field,” Eur. Phys. J. D 68, 215 (2014).
[Crossref]

Z. Ebadi and B. Mirza, “Entanglement generation by electric field background,” Ann. Phys. 351, 363–381 (2014).
[Crossref]

2013 (4)

S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A. Furusawa, “Deterministic quantum teleportation of photonic quantum bits by a hybrid technique,” Nature 500, 315–318 (2013).
[Crossref]

L. Memarzadeh and S. Mancini, “Entanglement dynamics for qubits dissipating into a common environment,” Phys. Rev. A 87, 032303 (2013).
[Crossref]

H. Katsuki, J. Delagnes, K. Hosaka, K. Ishioka, H. Chiba, E. Zijlstra, M. Garcia, H. Takahashi, K. Watanabe, M. Kitajima, Y. Matsumoto, K. G. Nakamura, and K. Ohmori, “All-optical control and visualization of ultrafast two-dimensional atomic motions in a single crystal of bismuth,” Nat. Commun. 4, 2801 (2013).
[Crossref]

N. B. An, “Protecting entanglement of atoms stored in a common nonperfect cavity without measurements,” Phys. Lett. A 377, 2520–2523 (2013).
[Crossref]

2012 (3)

A. Z. Chaudhry and J. Gong, “Decoherence control: universal protection of two-qubit states and two-qubit gates using continuous driving fields,” Phys. Rev. A 85, 012315 (2012).
[Crossref]

H.-S. Xu and J. B. Xu, “Protecting quantum correlations of two qubits in independent non-Markovian environments by bang-bang pulses,” J. Opt. Soc. Am. B 29, 2074–2079 (2012).
[Crossref]

Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nat. Phys. 8, 117–120 (2012).
[Crossref]

2011 (4)

F. Benatti and A. Nagy, “Three qubits in a symmetric environment: dissipatively generated asymptotic entanglement,” Ann. Phys. 326, 740–753 (2011).
[Crossref]

C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B 83, 115303 (2011).
[Crossref]

N. B. An, J. Kim, and K. Kim, “Entanglement dynamics of three interacting two-level atoms within a common structured environment,” Phys. Rev. A 84, 022329 (2011).
[Crossref]

J. M. Gambetta, A. A. Houck, and A. Blais, “Superconducting qubit with Purcell protection and tunable coupling,” Phys. Rev. Lett. 106, 030502 (2011).
[Crossref]

2010 (2)

S. Perseguers, M. Lewenstein, A. Acín, and J. I. Cirac, “Quantum random networks,” Nat. Phys. 6, 539–543 (2010).
[Crossref]

J.-Z. Hu, X.-B. Wang, and L. C. Kwek, “Protecting two-qubit quantum states by π-phase pulses,” Phys. Rev. A 82, 062317 (2010).
[Crossref]

2009 (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

2008 (1)

S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008).
[Crossref]

2007 (2)

T. Richter and W. Vogel, “Nonclassical characteristic functions for highly sensitive measurements,” Phys. Rev. A 76, 053835 (2007).
[Crossref]

B. Bellomo, R. Lo Franco, and G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[Crossref]

2005 (1)

D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. M. Girvin, and R. J. Schoelkopf, “AC Stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field,” Phys. Rev. Lett. 94, 123602 (2005).
[Crossref]

2004 (1)

J. K. Asbóth, P. Domokos, and H. Ritsch, “Correlated motion of two atoms trapped in a single-mode cavity field,” Phys. Rev. A 70, 013414 (2004).
[Crossref]

2001 (1)

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001).
[Crossref]

1999 (1)

M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, “Quantum telecloning and multiparticle entanglement,” Phys. Rev. A 59, 156–161 (1999).
[Crossref]

1998 (2)

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

1996 (1)

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996).
[Crossref]

1995 (1)

S. L. Braunstein and A. Mann, “Measurement of the Bell operator and quantum teleportation,” Phys. Rev. A 51, R1727–R1730 (1995).
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A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
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Figures (13)

Fig. 1.
Fig. 1. Pictorial representation of a setup where two qubits are moving inside a cavity. The qubits are two-level atoms with transition frequency ${\omega_0}$ travelling with constant velocity $v$.
Fig. 2.
Fig. 2. Stationary concurrence as a function of the relative coupling constant ${r_1}$ and the initial separability parameter $s$ for (a) $\varphi=0$ and (b) $\varphi=\pi$.
Fig. 3.
Fig. 3. Concurrence of motionless qubits ($\beta=0$) as a function of scaled time $\tau$ for $\varphi=0$ in the bad cavity limit, i.e., $R=0.1$ (top plots) and good cavity limit, i.e., $R=10$ (bottom plots) with (a), (c) factorized initial state, $s=1$ (left plots) and (b), (d) entangled initial state, $s=0$ (right plots) with the cases (i) maximal stationary value, ${r_1}=0.87$ (solid line), (ii) symmetric coupling, ${r_1}=1/\sqrt 2$ (dashed line), and (iii) only one coupled atom, ${r_1}=0$ or ${r_1}=1$ (dotted-dashed line).
Fig. 4.
Fig. 4. Concurrence as a function of scaled time $\tau$ for $\varphi=0$ in the bad cavity limit, i.e., $R=0.1$ with (a) entangled initial state, $s=0$ and (b) factorized initial state, $s=1$. In these plots, we have set ${r_1}=0.87$, ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line).
Fig. 5.
Fig. 5. Concurrence as a function of scaled time $\tau$ for $\varphi=0$ in the good cavity limit, i.e., $R=10$ with (a) entangled initial state, $s=0$ and (b) factorized initial state, $s=1$. In these plots, we have set ${r_1}=0.87$, ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line).
Fig. 6.
Fig. 6. Pairwise concurrence ${{\cal C}_{{\rm pair}}}$ as a function of $\tau$ for an initially $W$ state, for (a) weak coupling regime, $R=0.1$ and (b) strong coupling regime, $R=10$ with $n=2$ (solid black line), $n=4$ (dashed red line), and $n=8$ (dotted-dashed green line).
Fig. 7.
Fig. 7. Pairwise concurrence ${{\cal C}_{{\rm pair}}}$ as a function of $\tau$ for an initially $W$ state for (a) weak coupling regime, $R=0.1$ and (b) strong coupling regime, $R=10$. In these plots, we have set $n=4$, ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line).
Fig. 8.
Fig. 8. Pairwise concurrence ${{\cal C}_{j,l}}$ as a function of scaled time $\tau$ for $s=0$ and $\varphi=0$ with zero velocity for (a) bad cavity limit, $R=0.1$ and (b) good cavity limit, $R=10$ with $n=2$ (solid black lines), $n=6$ (dashed red lines), and $n=12$ (dotted-dashed green lines).
Fig. 9.
Fig. 9. ${{\cal C}_{j,l}}$ as a function of $\tau$ for system size $n=6$ for (a) bad cavity limit, $R=0.1$ (left plots) and (b) good cavity limit, $R=10$ with ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line). Other parameters are similar to Fig. 8.
Fig. 10.
Fig. 10. ${{\cal C}_{j,m}}$ as a function of $\tau$ for system size $n=6$ for (a) bad cavity limit, $R=0.1$ and (b) good cavity limit, $R=10$ with ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line). Other parameters are similar to Fig. 8.
Fig. 11.
Fig. 11. ${{\cal C}_{k,m}}$ as a function of $\tau$ for system size $n=4$ for (a) bad cavity limit, $R=0.1$ and (b) good cavity limit, $R=10$ with ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line). Other parameters are similar to Fig. 8.
Fig. 12.
Fig. 12. ${{\cal C}_{l,m}}$ as a function of $\tau$ for system size $n=4$ and $s=-1$ for (a) bad cavity limit, $R=0.1$ and (b) good cavity limit, $R=10$ (right plots) with ${\omega_0}/\lambda=1.5\times {10^9}$ and (i) $\beta=0$ (solid line), (ii) $\beta=2\times {10^{-9}}$ (dashed line), and (iii) $\beta=4\times {10^{-9}}$ (dotted-dashed line).
Fig. 13.
Fig. 13. Pictorial representation of the leading stationary concurrence when (a) the system is initially in a maximally entangled state of two qubits, i.e., $s=0$ and (b) initially one qubit is excited, i.e., $s=+1$. Blue (red) circle represents the qubits initially in the ground state (excited state). The lines between the circles indicate the correlation between them at steady state. The thicker the lines between two circles, the more stationary entanglement between them.

Equations (63)

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H ^ = H ^ 0 + H ^ i n t ,
H ^ 0 = i = 1 2 ω i σ ^ + ( i ) σ ^ ( i ) + k ω k a ^ k a ^ k
H ^ i n t = i = 1 2 k α i σ ^ + ( i ) g k f k i ( z ) a ^ k + H . c .
f k ( z i ) = f k ( v i t ) = sin [ ω k ( β i t Γ ) ] i = 1 , 2 ,
| ψ 0 = ( c 01 | e , g + c 02 | g , e ) | 0 R ,
| ψ ( t ) = c 1 ( t ) e i ω 1 t | e , g | 0 R + c 2 ( t ) e i ω 2 t | g , e | 0 R + k c k ( t ) e i ω k t | g , g | 1 k ,
ρ ^ ( t ) = ( 0 0 0 0 0 | c 1 ( t ) | 2 c 1 ( t ) c 2 ( t ) 0 0 c 1 ( t ) c 2 ( t ) | c 2 ( t ) | 2 0 0 0 0 1 | c 1 ( t ) | 2 | c 2 ( t ) | 2 ) .
c ˙ j ( t ) = i α j k g k f k ( z j ) c k ( t ) e i δ k ( j ) t , j = 1 , 2 ,
c ˙ k ( t ) = i g k j = 1 2 α j f k ( z j ) c j ( t ) e i δ k ( j ) t ,
c ˙ 1 ( t ) = 0 t k | g k | 2 e i δ k ( t t ) ( α 1 2 f k ( v 1 t ) f k ( v 1 t ) c 1 ( t ) + α 1 α 2 f k ( v 1 t ) f k ( v 2 t ) c 2 ( t ) ) d t , c ˙ 2 ( t ) = 0 t k | g k | 2 e i δ k ( t t ) ( α 2 2 f k ( v 2 t ) f k ( v 2 t ) c 2 ( t ) + α 1 α 2 f k ( v 2 t ) f k ( v 1 t ) c 1 ( t ) ) d t .
c ˙ 1 ( t ) = 0 t F ( t , t ) ( α 1 2 c 1 ( t ) + α 1 α 2 c 2 ( t ) ) d t ,
c ˙ 2 ( t ) = 0 t F ( t , t ) ( α 2 2 c 2 ( t ) + α 1 α 2 c 1 ( t ) ) d t ,
F ( t , t ) = k | g k | 2 e i δ k ( t t ) f k ( v t ) f k ( v t ) .
| ψ = r 2 | e , g r 1 | g , e ,
| ψ + = r 1 | e , g + r 2 | g , e .
Q ˙ ( t ) = α T 2 0 t F ( t , t ) Q ( t ) d t .
F ( t , t ) = W 2 λ π d ω sin [ ω ( β t Γ ) ] sin [ ω ( β t Γ ) ] ( ω ω 0 ) 2 + λ 2 × e i ( ω ω 0 ) ( t t ) .
F ( t , t ) = W 2 2 e λ ( t t ) cosh [ β λ ¯ ( t t ) ] ,
Q ( t ) = ( q 1 + y + ) ( q 1 + y ) ( q 1 q 2 ) ( q 1 q 3 ) e q 1 λ t + ( q 2 + y + ) ( q 2 + y ) ( q 2 q 1 ) ( q 2 q 3 ) e q 2 λ t + ( q 3 + y + ) ( q 3 + y ) ( q 3 q 1 ) ( q 3 q 2 ) e q 3 λ t ,
q 3 + 2 q 2 + ( y + y + R 2 2 ) q + R 2 2 = 0 ,
c 1 ( t ) = r 2 β + r 1 Q ( t ) β + ,
c 2 ( t ) = r 1 β + r 2 Q ( t ) β + .
C ( t ) := max { 0 , 1 2 3 4 } ,
C ( t ) = 2 | c 1 ( t ) c 2 ( t ) | .
C s = 2 | r 1 r 2 | | β | 2 .
c 01 = 1 s 2 ,
c 02 = 1 + s 2 e i φ ,
H ^ = ω 0 i = 1 n ( σ ^ + ( i ) σ ^ ( i ) ) + k ω k a ^ k a ^ k + α T i = 1 n k σ ^ + ( i ) g k f k ( z ) a ^ k + H . c . ,
| ψ ( 0 ) = | W | 0 R ,
| ψ ( t ) = D ( t ) e i ω 0 t | W | 0 R + k Λ k ( t ) e i ω k t | 1 k | G ,
| D ( t ) | 2 P 0 ( t ) = | ψ ( 0 ) | ψ ( t ) | 2 .
D ˙ ( t ) = n α T 2 0 t F ( t , t ) D ( t ) d t ,
q 3 + 2 q 2 + ( y + y + n R 2 2 ) q + n R 2 2 = 0.
ρ p a i r ( t ) = 1 n ( 0 0 0 0 0 | D ( t ) | 2 | D ( t ) | 2 0 0 | D ( t ) | 2 | D ( t ) | 2 0 0 0 0 n 2 | D ( t ) | 2 ) .
| ψ ( 0 ) = ( c 01 | e j + c 02 | e l ) | 0 R ,
| ψ ( t ) = ( C 1 ( t ) | e j + C 2 ( t ) | e l + C 3 ( t ) | E jl ) e i ω 0 t | 0 R + k C k ( t ) e i ω k t | 1 k | G ,
C 1 ( t ) = ( n 1 ) c 01 c 02 n + c 01 + c 02 n D ( t ) ,
C 2 ( t ) = ( n 1 ) c 02 c 01 n + c 01 + c 02 n D ( t ) ,
C 3 ( t ) = n 2 n ( c 01 + c 02 ) ( 1 + D ( t ) ) ,
ρ ( t ) = | C 1 ( t ) | 2 | e j e j | + | C 2 ( t ) | 2 | e l e l | + | C 3 ( t ) | 2 | E / j / l E / j / l | + ( C 1 ( t ) C 2 ( t ) | e j e l | + C 1 ( t ) C 3 ( t ) | e j E / j / l | + C 2 ( t ) C 3 ( t ) | e l E / j / l | + H.c. ) + ( 1 | C 1 ( t ) | 2 | C 2 ( t ) | 2 | C 3 ( t ) | 2 ) | G G | .
ρ j , l ( t ) = | C 1 ( t ) | 2 | e , g e , g | + | C 2 ( t ) | 2 | g , e g , e | + C 1 ( t ) C 2 ( t ) | e , g g , e | + C 1 ( t ) C 2 ( t ) | g , e e , g | + ( 1 | C 1 ( t ) | 2 | C 2 ( t ) | 2 ) | g , g g , g | .
C j , l ( t ) = 2 | C 1 ( t ) | | C 2 ( t ) | .
C j , l ( ) = ( n 2 ) 2 n 2 .
C j , m ( t ) = 2 n 2 | C 1 ( t ) | | C 3 ( t ) | ,
C j , m ( ) = 2 ( n 2 ) n 2 .
C k , m ( t ) = 2 n 2 | C 3 ( t ) | 2 ,
C l , m ( ) = 2 ( n 1 ) n 2 .
| ψ + = r 1 | e , g + r 2 | g , e .
| ψ + ( t ) = c 1 ( t ) | e , g + c 2 ( t ) | g , e .
Q ( t ) ψ + | ψ + ( t ) = ( r 1 e , g | + r 2 g , e | ) ( c 1 ( t ) | e , g + c 2 ( t ) | g , e ) = r 1 c 1 ( t ) + r 2 c 2 ( t ) .
Q ˙ ( t ) = r 1 c ˙ 1 ( t ) + r 2 c ˙ 2 ( t ) .
Q ˙ ( t ) = 0 t r 1 F ( t , t ) ( α 1 2 c 1 ( t ) + α 1 α 2 c 2 ( t ) ) d t 0 t r 2 F ( t , t ) ( α 2 2 c 2 ( t ) + α 1 α 2 c 1 ( t ) ) d t ,
Q ˙ ( t ) = 0 t F ( t , t ) ( c 1 ( t ) ( r 1 α 1 2 + r 2 α 1 α 2 ) + c 2 ( t ) ( r 1 α 1 α 2 + r 2 α 2 2 ) d t ,
Q ˙ ( t ) = α T 2 0 t F ( t , t ) ( r 1 c 1 ( t ) + r 2 c 2 ( t ) ) d t .
Q ˙ ( t ) = α T 2 0 t F ( t , t ) Q ( t ) d t .
Q ˙ ( t ) = α T 2 ( Q F ) ( t ) ,
s Q ( s ) 1 = α T 2 Q ( s ) F ( s ) ,
G ( s ) = W 2 4 ( 1 s + v + + 1 s + v ) ,
Q ( s ) = ( s + v + ) ( s + v ) ( s s 1 ) ( s s 2 ) ( s s 3 ) ,
s 3 + 2 λ s 2 + ( v + v + ( α T W ) 2 2 ) s + ( α T W ) 2 2 λ = 0.
q s λ , y ± v ± λ , R = α T W , R R λ ,
q 3 + 2 q 2 + ( y + y + R 2 2 ) q + R 2 2 = 0.
Q ( t ) = ( q 1 + y + ) ( q 1 + y ) ( q 1 q 2 ) ( q 1 q 3 ) e q 1 λ t + ( q 2 + y + ) ( q 2 + y ) ( q 2 q 1 ) ( q 2 q 3 ) e q 2 λ t + ( q 3 + y + ) ( q 3 + y ) ( q 3 q 1 ) ( q 3 q 2 ) e q 3 λ t .

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