Abstract

A precise time-dependent control of a quantum system relies on an accurate account of the quantum interference among the system, the control, and the environment. A diagrammatic technique has been recently developed to precisely calculate this quantum correlation for a fast multimode coherent photon control against slow relaxation, valid for both Markovian and non-Markovian systems. We review this formalism in comparison with the existing approximate theories and extend it to cases with controls by photon state other than the coherent state.

© 2012 Optical Society of America

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  1. P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
    [CrossRef]
  2. C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2000).
  3. P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics, (Princeton University, 2011).
  4. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455–503 (2000).
    [CrossRef]
  5. P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
    [CrossRef]
  6. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
    [CrossRef]
  7. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
    [CrossRef]
  8. C. K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
    [CrossRef]
  9. L. V. Keldysh, “Diagram technique for non-equilibrium processes,” JETP 20, 1018–1027 (1965).
  10. G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80, 268–272 (1950).
    [CrossRef]
  11. G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
    [CrossRef]
  12. H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
    [CrossRef]
  13. H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University, 2002).
  14. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  15. G. J. Milburn, “Interaction of a two-level atom with squeezed light,” Optica Acta 31, 671–679 (1984).
    [CrossRef]

2011

C. K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
[CrossRef]

2009

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

2004

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

2000

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef]

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

1999

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
[CrossRef]

1984

G. J. Milburn, “Interaction of a two-level atom with squeezed light,” Optica Acta 31, 671–679 (1984).
[CrossRef]

1965

L. V. Keldysh, “Diagram technique for non-equilibrium processes,” JETP 20, 1018–1027 (1965).

1950

G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80, 268–272 (1950).
[CrossRef]

Aliferis, P.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Bay, S.

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

Berman, P. R.

P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics, (Princeton University, 2011).

Breuer, H.-P.

H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
[CrossRef]

H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University, 2002).

Brinati, J. R.

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

Brito, F.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Chan, C. K.

C. K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
[CrossRef]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef]

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

DiVincenzo, D. P.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Dodonov, V. V.

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Gardiner, C.

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2000).

Gibbs, H. M.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Imada, M.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef]

Irman, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Kappler, B.

H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
[CrossRef]

Keldysh, L. V.

L. V. Keldysh, “Diagram technique for non-equilibrium processes,” JETP 20, 1018–1027 (1965).

Khitrova, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Lambropoulos, P.

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

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Malinovsky, V. S.

P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics, (Princeton University, 2011).

Milburn, G. J.

G. J. Milburn, “Interaction of a two-level atom with squeezed light,” Optica Acta 31, 671–679 (1984).
[CrossRef]

Mizrahi, S. S.

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

Nielsen, T. R.

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

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Nikolopoulos, G. M.

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

Noda, S.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef]

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Petruccione, F.

H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
[CrossRef]

H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University, 2002).

Prataviera, G. A.

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

Preskill, J.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Rupper, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Scherer, A.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

Sham, L. J.

C. K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
[CrossRef]

Shchekin, O. B.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Steffen, M.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Terhal, B. M.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Wick, G. C.

G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80, 268–272 (1950).
[CrossRef]

Yoshie, T.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Zoller, P.

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2000).

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

JETP

L. V. Keldysh, “Diagram technique for non-equilibrium processes,” JETP 20, 1018–1027 (1965).

Nature

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
[CrossRef]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

New J. Phys.

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11, 013061 (2009).
[CrossRef]

Optica Acta

G. J. Milburn, “Interaction of a two-level atom with squeezed light,” Optica Acta 31, 671–679 (1984).
[CrossRef]

Phys. Rev.

G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80, 268–272 (1950).
[CrossRef]

Phys. Rev. A

G. A. Prataviera, S. S. Mizrahi, V. V. Dodonov, and J. R. Brinati, “Probing colored noise from the index of refraction of strongly driven two-level atoms,” Phys. Rev. A 60, 4045–4051 (1999).
[CrossRef]

H.-P. Breuer, B. Kappler, and F. Petruccione, “Stochastic wave-function method for non-Markovian quantum master equations,” Phys. Rev. A 59, 1633–1643 (1999).
[CrossRef]

C. K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
[CrossRef]

Rep. Prog. Phys.

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

Other

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2000).

P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics, (Princeton University, 2011).

H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University, 2002).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

An illustration of the TLS-multiphoton process conditioned on a multimode coherent state. The ± sign labels the spin state of the TLS. Contractions among photons (dashed curve) correspond to decoherence, while the normal ordered control photons (wavy lines) is responsible for the Rabi motion of the TLS.

Fig. 2.
Fig. 2.

Diagrammatic representations of the Wick’s expansion of the transformation matrix psf,sf;s,s(t,{α}) for a coherent control. (a) Two possible contractions between photons at different times. (b) Two types of dressed lines that represent a sum of even or odd numbers of control photons interacting with the TLS. (c) Dressed diagram without any contraction represents the classical Rabi solution of p++;++ from time 0 to t. (d) Vacuum relaxation is represented by undressed diagrams with contractions only. (e) Leading contribution of control noise to p++;++ comes from three dressed diagrams with only one contraction.

Fig. 3.
Fig. 3.

A comparison between the diagrammatic methods and the ME approaches presented in [8], using a coherently driven single mode JC system with n¯=100π2, corresponding to a 4π rotation at gt=0.2. The magnitudes of errors of the ME approaches and the classical Rabi solution are comparable even in the small time regime.

Fig. 4.
Fig. 4.

Fifteen dressed diagrams with two contractions for p++;++ under a coherent control.

Fig. 5.
Fig. 5.

A plot of absolute errors of different methods under the same physical situation as in Fig. 3. Diagrammatic solution with n contractions has an error of O[(gt)2(n+1)].

Fig. 6.
Fig. 6.

Eight possible contractions between photons for a squeezed vacuum.

Fig. 7.
Fig. 7.

P++(t)P++classical(t) of an initially exited TLS under a single mode squeezed coherent state with n¯=100π2 for various squeezing parameters. ϕ is the phase of α* and θ is the phase of squeezing. The curves and symbols correspond to the diagrammatic and exact solutions, respectively.

Fig. 8.
Fig. 8.

Relaxation of an initially excited TLS under different initial photon states. N=|α|2=103. The case with a nondissipative single-mode number state (|N) gains decoherence when a degenerate vacuum mode is added (|N,0). In the small time region, the amount of dissipation is the same as that of a single-mode coherent state (|α).

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

U(t)=Texp[i0tdtV(t)],
V(tl)=σ+Al+σAl,
Al=kgkakei(ω0ωk)tl.
k(gkeiωktαk+c.c)=Ω(t)2ei(ω0t+ϕ)+c.c.,
psf,sf;s,s(t,α)=α|s|U(t)|sfsf|U(t)|s|α×ei(sf1sf1)ω0t/2,
Psf,sf(t)=s,scscs*psf,sf;s,s(t,α).
p++;++(t)=n,n=0(1)n+n0tD2nt0tD2nt×α|(A1A2A2n1A2n)(A2nA2n1A2A1)|α,
AiAj=K(titj)=kgk2ei(ω0ωk)(titj),
De(t,t)=cos(A(t)A(t)2)Θ(tt),Do(t,t)=(±i)(e±iϕ)sin(A(t)A(t)2)Θ(tt),
ddtρs(t)=i[Hc(t),ρs(t)]+0tdtL^(tt)ρs(t),
ddtρsNZ(t)=iTrR[V(t),ρsNZ(t)ρR]+0tdtTrR[V(t),TrR[V(t),ρsNZ(t)ρR]ρR]0tdtTrR[V(t),[V(t),ρsNZ(t)ρR]],
ddtρsNZ(t)=iΩ(t)2[σ+eiϕ+σeiϕ,ρsNZ(t)]0tdtK(tt)[σ+σρsNZ(t)σρsNZ(t)σ++h.c.],
SakS=akcoshra2k¯keiθsinhr,SakS=akcoshra2k¯keiθsinhr,
A˜iA˜j=cosh2rkgk2ei(ω0ωk)(titj),A˜iA˜j=sinh2rkgk2ei(ω0ωk)(titj),A˜iA˜j=eiθ2sinh(2r)×kgkg2k¯kei(ω0ωk)tiei(ω0ω2k¯k)tj,A˜iA˜j=A˜iA˜j*,
p++;++(0)(t)=1N(gt)2+n,n=1(1)n+n(gt)2n+2n(2n)!(2n)!×N(N1)(Nnn+1),

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