Abstract

We propose a formulation to obtain the exact susceptibility for system arbitrary operators to the external fields by means of the whole-system Hamiltonian (system plus reservoir) diagonalization methods, where the dissipative effects directly reflect the nature of the structured non-Markovian reservoir. This treatment does not make the Born-Markovian approximation in structured non-Markovian reservoir. The relations between linear response function and bound-states for the system as well as structured reservoir are found, which shows the photon bound-states and continuous energy spectrum can be readout from the susceptibility, respectively. These results are then used to examine the validity of second-order Born-Markovian approximation, where we find interesting features (e.g., bound-states) are lost in the approximate treatments for open systems. We study the dependence of the response function on the type (spectrum density) of interaction between the system and structured reservoir. We also give the physical reasons behind the disappearance of the bound-states in the approximation method. Finally, these results are also extended to a more general quantum network involving an arbitrary number of coupled-bosonic system without rotating-wave approximation. The presented results might open a new door to understand the linear response and the energy spectrum for non-Markovian open systems with structured reservoirs.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

H. Z. Shen, S. Xu, H. T. Cui, and X. X. Yi, “Non-Markovian dynamics of a system of two-level atoms coupled to a structured environment,” Phys. Rev. A 99(3), 032101 (2019).
[Crossref]

2018 (6)

H. Z. Shen, S. Xu, S. Yi, and X. X. Yi, “Controllable dissipation of a qubit coupled to an engineering reservoir,” Phys. Rev. A 98(6), 062106 (2018).
[Crossref]

H. T. Cui, H. Z. Shen, S. C. Hou, and X. X. Yi, “Bound state and localization of excitation in many-body open systems,” Phys. Rev. A 97(4), 042129 (2018).
[Crossref]

C. E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, and S. Diehl, “Probing the Topology of Density Matrices,” Phys. Rev. X 8(1), 011035 (2018).
[Crossref]

H. Z Shen, S. Xu, H. Li, S. L. Wu, and X. X. Yi, “Linear response theory for periodically driven systems with non-Markovian effects,” Opt. Lett. 43(12), 2852 (2018).
[Crossref]

T. Ozawa, “Steady-state Hall response and quantum geometry of driven-dissipative lattices,” Phys. Rev. B 97(4), 041108 (2018).
[Crossref]

S. Bachmann, W. De Roeck, and M. Fraas, “The adiabatic theorem and linear response theory for extended quantum systems,” Commun. Math. Phys. 361(3), 997–1027 (2018).
[Crossref]

2017 (11)

Y. Liu and A. A. Houck, “Quantum electrodynamics near a photonic bandgap,” Nat. Phys. 13(1), 48–52 (2017).
[Crossref]

G. Diezemann, “Nonlinear response theory for Markov processes. II. Fifth-order response functions,” Phys. Rev. E 96(2), 022150 (2017).
[Crossref]

M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, “Geometry and non-adiabatic response in quantum and classical systems,” Phys. Rep. 697, 1–87 (2017).
[Crossref]

P. Weinberg, M. Bukov, L. D’Alessio, A. Polkovnikov, S. Vajna, and M. Kolodrubetz, “Adiabatic perturbation theory and geometry of periodically-driven systems,” Phys. Rep. 688, 1–35 (2017).
[Crossref]

M. Ban, S. Kitajima, T. Arimitsu, and F. Shibata, “Linear response theory for open systems: quantum master equation approach,” Phys. Rev. A 95(2), 022126 (2017).
[Crossref]

H. Z. Shen, D. X. Li, and X. X. Yi, “Non-Markovian linear response theory for quantum open systems and its applications,” Phys. Rev. E 95(1), 012156 (2017).
[Crossref]

H. Z. Shen, H. Li, Y. F. Peng, and X. X. Yi, “Mechanism for Hall conductance of two-band systems against decoherence,” Phys. Rev. E 95(4), 042129 (2017).
[Crossref]

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89(1), 015001 (2017).
[Crossref]

G. Calajó and Peter Rabl, “Strong coupling between moving atoms and slow-light Cherenkov photons,” Phys. Rev. A 95(4), 043824 (2017).
[Crossref]

A. González-Tudela and J. I. Cirac, “Markovian and non-Markovian dynamics of quantum emitters coupled to two-dimensional structured reservoirs,” Phys. Rev. A 96(4), 043811 (2017).
[Crossref]

A. González-Tudela and J. I. Cirac, “Quantum Emitters in Two-Dimensional Structured Reservoirs in the Nonperturbative Regime,” Phys. Rev. Lett. 119(14), 143602 (2017).
[Crossref]

2016 (9)

T. Shi, Y. H. Wu, A. González-Tudela, and J. I. Cirac, “Bound States in Boson Impurity Models,” Phys. Rev. X 6(2), 021027 (2016).
[Crossref]

G. Calajó, F. Ciccarello, D. Chang, and P. Rabl, “Atom-field dressed states in slow-light waveguide QED,” Phys. Rev. A 93(3), 033833 (2016).
[Crossref]

L. Ferialdi, “Exact Closed Master Equation for Gaussian Non-Markovian Dynamics,” Phys. Rev. Lett. 116(12), 120402 (2016).
[Crossref]

H. P. Breuer, E. M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-Markovian dynamics in open quantum systems,” Rev. Mod. Phys. 88(2), 021002 (2016).
[Crossref]

H. Z. Shen, X. Q. Shao, G. C. Wang, X. L. Zhao, and X. X. Yi, “Quantum phase transition in a coupled two-level system embedded in anisotropic three-dimensional photonic crystals,” Phys. Rev. E 93(1), 012107 (2016).
[Crossref]

Z. C. Shi, H. Z. Shen, W. Wang, and X. X. Yi, “Response of two-band systems to a single-mode quantized field,” Phys. Rev. E 93(3), 032120 (2016).
[Crossref]

T. Shitara and M. Ueda, “Determining the continuous family of quantum Fisher information from linear-response theory,” Phys. Rev. A 94(6), 062316 (2016).
[Crossref]

L. C. Venuti and P. Zanardi, “Dynamical response theory for driven-dissipative quantum systems,” Phys. Rev. A 93(3), 032101 (2016).
[Crossref]

V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, “Geometry and response of Lindbladians,” Phys. Rev. X 6(4), 041031 (2016).
[Crossref]

2015 (5)

H. Z. Shen, W. Wang, and X. X. Yi, “Hall conductance and topological invariant for open systems,” Sci. Rep. 4(1), 6455 (2015).
[Crossref]

F. Sakuldee and S. Suwanna, “Linear response and modified fluctuation-dissipation relation in random potential,” Phys. Rev. E 92(5), 052118 (2015).
[Crossref]

H. Z. Shen, M. Qin, X. Q. Shao, and X. X. Yi, “General response formula and application to topological insulator in quantum open system,” Phys. Rev. E 92(5), 052122 (2015).
[Crossref]

M. Ban, “Linear response of a pre-and post-selected system to an external field,” Phys. Lett. A 379(4), 284–288 (2015).
[Crossref]

M. Ban, “Linear response theory for open quantum systems within the framework of the ABL formalism,” Quantum Stud.: Math. Found. 2(1), 51–62 (2015).
[Crossref]

2014 (3)

L. Diósi and L. Ferialdi, “General Non-Markovian Structure of Gaussian Master and Stochastic Schrödinger Equations,” Phys. Rev. Lett. 113(20), 200403 (2014).
[Crossref]

C. J. Yang, J. H. An, H. G. Luo, Y. d. Li, and C. H. Oh, “Canonical versus noncanonical equilibration dynamics of open quantum systems,” Phys. Rev. E 90(2), 022122 (2014).
[Crossref]

C. Y. Cai, L. P. Yang, and C. P. Sun, “Threshold for nonthermal stabilization of open quantum systems,” Phys. Rev. A 89(1), 012128 (2014).
[Crossref]

2013 (1)

A. J. van Wonderen and L. G. Suttorp, “Kraus map for non-Markovian quantum dynamics driven by a thermal reservoir,” Europhys. Lett. 102(6), 60001 (2013).
[Crossref]

2012 (4)

W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W. Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109(17), 170402 (2012).
[Crossref]

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J. E. Avron, M. Fraas, and G. M. Graf, “Adiabatic response for Lindblad dynamics,” J. Stat. Phys. 148(5), 800–823 (2012).
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2011 (5)

A. R. Kolovsky, “Hall conductivity beyond the linear response regime,” Europhys. Lett. 96(5), 50002 (2011).
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J. E. Avron, M. Fraas, G. M. Graf, and O. Kenneth, “Quantum response of dephasing open systems,” New J. Phys. 13(5), 053042 (2011).
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2010 (4)

A. Smirne and B. Vacchini, “Nakajima-Zwanzig versus time-convolutionless master equation for the non-Markovian dynamics of a two-level system,” Phys. Rev. A 82(2), 022110 (2010).
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M. H. Wu, C. U. Lei, W. M. Zhang, and H. N. Xiong, “Non-Markovian dynamics of a microcavity coupled to a waveguide in photonic crystals,” Opt. Express 18(17), 18407–18418 (2010).
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2009 (4)

H. Jiang, S. G. Cheng, Q. F. Sun, and X. C. Xie, “Topological Insulator: A New Quantized Spin Hall Resistance Robust to Dephasing,” Phys. Rev. Lett. 103(3), 036803 (2009).
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C. Uchiyama, M. Aihara, M. Saeki, and S. Miyashita, “Master equation approach to line shape in dissipative systems,” Phys. Rev. E 80(2), 021128 (2009).
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S. Longhi, “Spectral singularities in a non-Hermitian Friedrichs-Fano-Anderson model,” Phys. Rev. B 80(16), 165125 (2009).
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2008 (4)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
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H. G. Park, C. J. Barrelet, Y. Wu, B. Tian, F. Qian, and C. M. Lieber, “A wavelength-selective photonic-crystal waveguide coupled to a nanowire light source,” Nat. Photonics 2(10), 622–626 (2008).
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F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay Control via Discrete-to-Continuum Coupling Modulation in an Optical Waveguide System,” Phys. Rev. Lett. 101(14), 143602 (2008).
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I. de Vega, D. Porras, and J. Ignacio Cirac, “Matter-WaveEmission in Optical Lattices: Single Particle and Collective Effects,” Phys. Rev. Lett. 101(26), 260404 (2008).
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2007 (1)

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
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2006 (2)

X. L. Qi, Y. S. Wu, and S. C. Zhang, “Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors,” Phys. Rev. B 74(8), 085308 (2006).
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B. Zhou, L. Ren, and S. Q. Shen, “Spin transverse force and intrinsic quantum transverse transport,” Phys. Rev. B 73(16), 165303 (2006).
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2005 (1)

2004 (2)

P. Lodahl, A. F. Van Driel, I. N. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature (London) 430(7000), 654–657 (2004).
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J. K. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665 (2004).
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2001 (1)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87(25), 253902 (2001).
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2000 (1)

P. Lambropoulos, G. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63(4), 455–503 (2000).
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1999 (2)

H. P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59(2), 1633–1643 (1999).
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A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711 (1999).
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1996 (1)

T. S. Evans and D. A. Steer, “Wick’s theorem at finite temperature,” Nucl. Phys. B 474(2), 481–496 (1996).
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1994 (1)

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50(2), 1764–1769 (1994).
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1991 (1)

S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43(16), 12772–12789 (1991).
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1990 (1)

S. John and J. Wang, “Quantum Electrodynamics near a Photonic Band Gap: Photon Bound States and Dressed Atoms,” Phys. Rev. Lett. 64(20), 2418–2421 (1990).
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1987 (1)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
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1985 (1)

M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160(2), 343–354 (1985).
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1982 (1)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405–408 (1982).
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1961 (2)

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1958 (1)

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S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
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H. Z. Shen, S. Xu, H. T. Cui, and X. X. Yi, “Non-Markovian dynamics of a system of two-level atoms coupled to a structured environment,” Phys. Rev. A 99(3), 032101 (2019).
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Figures (8)

Fig. 1.
Fig. 1. (a) A schematic plot of a single-mode microcavity coupled to a CROW structure. Microcavity field to the ${P}$-coupled-resonator at the first site in the waveguide with the coupling strength $\gamma$ which is also controllable experimentally by adjusting the distance between defects, and the photon hopping between two consecutive resonators in the waveguide structure with the controllable hopping amplitude $\lambda _0$. The cavity is small perturbed. (b) Dispersion relation in $k$ space for the system. (c) Spectrum density for the optical waveguide structure with $\omega _0=\lambda _0=\omega _c$, $\eta =1$.
Fig. 2.
Fig. 2. Imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with different system-reservoir coupling strength $\eta$. Here and hereafter, $\omega _0$, $\lambda _0$, and external field frequency $\omega$ are rescaled in units of $\omega _c$, and the time $t$ is then in units of $1/\omega _c$. Hence all parameters are dimensionless. The width $L_1-L_4$ emphasizes the width related to the continuous spectrum in the reservoir, while the points $A-D$ mark the positions of system-reservoir bound-states, respectively. Other parameter is $\lambda _0 = 0.5\omega _c$.
Fig. 3.
Fig. 3. Structure of solutions of eigenEq. (8). (a) Below the critical coupling, no solution exists outside the energy band. (b) Over the critical coupling, two solutions exist outside the energy band. (c) Critical regimes for different types of bound-states based on Eq. (33). In the regime $R_1$, no bound-states; in the regimes $R_2$ or $R_3$, one bound state; in regime $R_4$, two bound-states.
Fig. 4.
Fig. 4. The distribution of numerically calculated eigenenergy spectrum of Hamiltonian (1) at optical waveguide for $\lambda _0 = 0.5\omega _c$. Eigenenergies are arranged in asscending order, where the $k$ space is discretize into $101$ Bloch modes from $0$ to $\pi$, i.e., $k = 0,\pi /100, \cdot \cdot \cdot \pi$ in $k$ space. The parameters of Fig. 4 are the same as those in Fig. 2, respectively. The bulk of the spectrum remains fixed (see $L_1-L_4$), whereas the bound-state eigenenergies are outside the boundaries (see points $A-D$).
Fig. 5.
Fig. 5. The absolute value of microcavity field amplitude $|{\cal M}(t)|$ of the microcavity in photonic waveguide. The parameters chosen in Fig. 5 are the same as those in Fig. 2, respectively.
Fig. 6.
Fig. 6. Imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with different system-reservoir coupling strength $\eta$ and external field frequency $\omega$. The parameters chosen are $\lambda _0 = 0.5\omega _c$, $\omega _0 = \omega _c$ for (a), $\omega _0 = 1.5\omega _c$ for (b), $\omega _0 = 0.5\omega _c$ for (c) , where the $k$ space is discretize into $101$ Bloch modes from $0$ to $\pi$, i.e., $k = 0,\pi /100, \cdot \cdot \cdot \pi$) in $k$ space.
Fig. 7.
Fig. 7. Comparison of three methods for imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with the external field applied to the system. The red-solid line stands for the exact linear response (34), blue-dotted line for the linear response in the second-order Born approximation (39), and yellow-dashed line for the linear response obtained by applying Markovian limit (40). In this figure, we set $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.01$ for (a); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.05$ for (b); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.1$ for (c); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.5$ for (d); $\lambda _0 = 0.5\omega _c$, $\omega _0 = 1.5\omega _c$, $\eta =1.5$ for (e); $\lambda _0 = 0.5\omega _c$, $\omega _0 = 1.5\omega _c$, $\eta =4$ for (f).
Fig. 8.
Fig. 8. The figure shows that the response functions can be readout by probed the susceptibility of quantum many-body system consisting of $M$ mutually coupled charged-Brownian particles ($R_{mn}$ denotes the coupling matrix) interacting with the reservoir described by harmonic oscillators with frequency $\omega _k$. The large-circle denotes the charged-Brownian oscillator, which is interacted with a large number of oscillators (the reservoir shown by small circles) coupling by interacting strengths $v_{n,k}$.

Equations (45)

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H ^ = ω c a ^ a ^ + k ω k b ^ k b ^ k + k v k ( a ^ b ^ k + b ^ k a ^ ) ,
H ^ = j E j A ^ j A ^ j ,
A ^ j = α j a ^ + k β j ( k ) b ^ k ,
a ^ = j α j A ^ j , b ^ k = j β j ( k ) A ^ j .
k v k β j ( k ) ( E j ω c ) α j ,
v k α j β j ( k ) ( E j ω k ) ,
α B = [ 1 + k v k 2 ( E B ω k ) 2 ] 1 / 2 .
Y ( E B ) ω c + d Ω 2 π J ( Ω ) E B Ω E B ,
J ( Ω ) = 2 π k | v k | 2 δ ( Ω ω k ) .
ω c J ( ω ) ω d ω .
H ^ E j ω k = ( ω c v k S v k L v k S ω k S 0 0 0 0 v k L 0 0 ω k L ) ,
χ s ^ z 1 z 2 a ^ ( ω ) = lim ε 0 + i 0 d t e i ω t ε t Tr { s ^ z 1 z 2 ( t ) [ a ^ , ρ e q ] } , = lim ε 0 + i 0 d t e i ω t ε t Tr { s ^ z 1 z 2 [ a ^ ( t ) , ρ e q ] } ,
s ^ z 1 z 2 a ^ z 1 a ^ z 2 .
a ^ ( t ) = M ( t ) a ^ ( 0 ) + C ^ ( t ) ,
d d t M ( t ) = i ω c M ( t ) 0 t d τ f ( t τ ) M ( τ ) ,
d d t C ^ ( t ) = i ω c C ^ ( t ) 0 t d τ f ( t τ ) C ^ ( τ ) i k v k b ^ k ( 0 ) e i ω k τ ,
f ( t ) = k | v k | 2 e i ω k t ,
C ^ ( t ) = i k v k b ^ k ( 0 ) 0 t d τ e i ω k τ M ( t τ ) .
M ( t ) = B α B e i E B t + 2 π J ( Ω ) e i Ω t d Ω 4 [ Ω ω c S ( Ω ) ] 2 + J 2 ( Ω ) ,
χ s ^ z 1 z 2 a ^ ( ω ) = z 2 i M ( ω ) Z m 1 , m z 1 , n 1 , n z 2 1 , α m 1 α m z 1 α n 1 α n z 2 1 × Tr { A ^ m 1 A ^ m z 1 A ^ n 1 A ^ n z 2 1 e β j E j A ^ j A ^ j } ,
χ s ^ 01 a ^ ( ω ) = i M ( ω ) ,
χ s ^ 12 a ^ ( ω ) = 2 i M ( ω ) j | α j | 2 N j ,
χ s ^ 23 a ^ ( ω ) = 6 i M ( ω ) j , k | α j | 2 | α k | 2 N j N k ,
M ( ω ) = lim ε 0 + i 0 e i ω t ε t M ( t ) d t ,
H ^ 0 = ω c a ^ a ^ + m = 1 P ω 0 b ^ m b ^ m + γ ( a ^ b ^ 1 + b ^ 1 a ^ ) m = 1 P 1 λ 0 ( b ^ m b ^ m + 1 + b ^ m + 1 b ^ m ) .
[ b ^ k , b ^ k ] = δ k k , k 1 π 0 π d k , 0 π sin ( n k ) sin ( m k ) d k = π 2 δ m n ,
ω k = ω 0 2 λ 0 cos k , v k = 2 π γ sin k .
J ( Ω ) = 4 γ 2 0 π sin 2 k δ ( Ω ω 0 + 2 λ 0 cos k ) d k .
J ( Ω ) = { η 2 4 λ 0 2 ( Ω ω 0 ) 2 | Ω ω 0 | 2 λ 0 , 0 | Ω ω 0 | > 2 λ 0 ,
α ± = ( η 2 2 ) 4 ( η 2 1 ) λ 0 2 + Δ 2 ± Δ η 2 2 ( η 2 1 ) 4 ( η 2 1 ) λ 0 2 + Δ 2 , E ± = 2 ω 0 ( η 2 1 ) + ( η 2 2 ) Δ 2 ( η 2 1 ) ± η 2 4 ( η 2 1 ) λ 0 2 + Δ 2 2 ( η 2 1 ) .
χ a ^ a ^ ( ω ) = B = ± α B i ( E B ω i ε ) + 2 π i J ( Ω ) ( Ω ω i ε ) 1 d Ω 4 [ Ω ω 0 S ( Ω ) ] 2 + J 2 ( Ω ) ,
| Φ j = α j | 1 , { 0 k } + k β j ( k ) | 0 , 1 k ,
H ^ | Φ j = E j | Φ j ,
η ± = 2 Δ λ 0 ,
χ a ^ a ^ ( ω ) = lim ε 0 + ω c ω + d Ω 2 π J ( Ω ) ω + i ε Ω ,
| M ± ( t ) | = | α ± | .
| M ( t ) | = α + 2 + α 2 + 2 α + α cos [ ( α + α ) t ] ,
ρ ˙ S O B A = i [ H ^ S , ρ S O B A ( t ) ] + 0 t d t { F ( t t ) × [ a ^ ( t t ) ρ S O B A ( t ) a ^ a ^ a ^ ( t t ) ρ S O B A ( t ) ] + H . c . } ,
d ρ M A ( t ) d t = i [ H ^ S , ρ M A ( t ) ] + η 2 λ 0 2 [ 2 a ^ ρ M A ( t ) a ^ { a ^ a ^ , ρ M A ( t ) } ] .
χ a ^ a ^ ( ω ) = lim ε 0 + ω c ω i ε [ 1 + 1 ω + i ε × d Ω 2 π J ( Ω ) ω + ω c Ω + i ε ] .
χ a ^ a ^ ( ω ) = lim ε 0 + ω + i ε ( ω c η 2 λ 0 ) .
H ^ = m , n = 1 M R m n a m a ^ n + k ω k b ^ k b ^ k + n , k v n , k ( a ^ n + a n ) ( b ^ k + b ^ k ) ,
a ^ j ( t ) t = i n = 1 M R j n a ^ n ( t ) i [ D ^ j ( t ) + D ^ j ( t ) ] m = 1 M 0 t d τ F j m ( t τ ) [ a ^ m ( τ ) + a ^ m ( τ ) ] ,
A ˙ ( t ) = i R A ( t ) 0 t d t F ( t τ ) [ A ( τ ) + B ( τ ) ] , B ˙ ( t ) = i R B ( t ) 0 t d t F ( t τ ) [ A ( τ ) + B ( τ ) ] , Q ^ ˙ ( t ) = i R Q ^ ( t ) 0 t d t F ( t τ ) [ Q ^ ( τ ) + Q ^ ( τ ) ] i D ^ ( t ) i D ^ ( t ) ,
χ m n ( ω ) = lim ε 0 + i 0 d t e i ω t ε t T r S R { a ^ m ( t ) [ a ^ n , ρ e q ] } .

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