Abstract

A slow monoelectronic excitation in a covalent crystal at the temperature T=0 is analyzed. The interaction with zero-point longitudinal acoustic phonons leads to the formation of a dressed electronic state at an energy level lower than that of the initial bare state. This aspect of the dressing process is described here by hypothesizing that the excess of energy is released with the emission of real phonons. Specifically, this paper considers the transition probability from the bare monoelectronic state to a dressed state of the electron accompanied by real phonons and a deformation field. The spectrum of the real phonons emitted during the electronic self-dressing is calculated by applying nonperturbative techniques based on the resolvent method. The present analysis indicates that the transition between the bare monoelectronic state and the asymptotic dressed state is accompanied, at the lowest order, by the appearance of two real phonons.

© 2011 Optical Society of America

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2011

B. Luo, J. Ye, and Y. Zhao, “Variational study of polaron dynamics with the Davydov Ansätze,” Phys. Status Solidi C 8, 70–73 (2011).
[CrossRef]

2010

Y.-Y. Zhang, Q.-H. Chen, and K.-L. Wang, “Quantum phase transition in the sub-Ohmic spin-boson model: an extended coherent-state approach,” Phys. Rev. B 81, 121105 (2010).
[CrossRef]

B. Luo, J. Ye, C. Guan, and Y. Zhao, “Validity of time-dependent trial states for the Holstein polaron,” Phys. Chem. Chem. Phys. 12, 15073–15084 (2010).
[CrossRef] [PubMed]

J. Sun, B. Luo, and Y. Zhao, “Dynamics of a one-dimensional Holstein polaron with the Davydov ansätze,” Phys. Rev. B 82, 014305 (2010).
[CrossRef]

2009

S. Mitsubori, I. Katayama, S. H. Lee, T. Yao, and J. Takeda, “Ultrafast lasing due to electron hole plasma in ZnO nano-multipods,” J. Phys. Condens. Matter 21, 064211 (2009).
[CrossRef] [PubMed]

2008

2007

P. Gaal, W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, “Internal motions of a quasiparticle governing its ultrafast nonlinear response,” Nature 450, 1210–1213 (2007).
[CrossRef] [PubMed]

L.-C. Ku and S. Trugman, “Quantum dynamics of polaron formation,” Phys. Rev. B 75, 014307 (2007).
[CrossRef]

D. Polli, L. Lüer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108–103116 (2007).
[CrossRef] [PubMed]

2006

B. Zheng, J. Wu, W. Sun, and C. Liu, “Trapping and hopping of polaron in DNA periodic stack,” Chem. Phys. Lett. 425, 123–127(2006).
[CrossRef]

X. Liu, K. Gao, J. Fu, Y. Li, J. Wei, and S. Xie, “Effect of the electric field mode on the dynamic process of a polaron,” Phys. Rev. B 74, 172301 (2006).
[CrossRef]

2005

2004

Z. An, C. Q. Wu, and X. Sun, “Dynamics of photogenerated polarons in conjugated polymers,” Phys. Rev. Lett. 93, 216407(2004).
[CrossRef] [PubMed]

2003

2002

G. Compagno and D. Valenti, “Radiative emission due to atomic self-dressing in QED,” Phys. Rev. A 65, 032106 (2002).
[CrossRef]

P. Gartner, L. Bányai, and H. Haug, “Self-consistent RPA for the intermediate-coupling polaron,” Phys. Rev. B 66, 075205(2002).
[CrossRef]

C. Gadermaier and G. Lanzani, “Photophysics of conjugated polymers: the contribution of ultrafast spectroscopy,” J. Phys. Condens. Matter 14, 9785–9802 (2002).
[CrossRef]

2001

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

2000

A. Zewail, “Femtochemistry: atomic-scale dynamics of the chemical bond,” J. Phys. Chem. A 104, 5660–5694 (2000).
[CrossRef]

T. Polívka, T. Pullerits, J. Herek, and V. Sundström, “Exciton relaxation and polaron formation in LH2 at low temperature,” J. Phys. Chem. B 104, 1088–1096 (2000).
[CrossRef]

D. Valenti and G. Compagno, “Self-dressing dynamics of slow electrons in covalent crystals,” AIP Conf. Proc. 513, 242–245(2000).
[CrossRef]

E. Karpov, I. Prigogine, T. Petrosky, and G. Pronko, “Friedrichs model with virtual transitions. Exact solution and indirect spectroscopy,” J. Math. Phys. 41, 118–131 (2000).
[CrossRef]

1999

P. Gartner, L. Bányai, and H. Haug, “Two-time electron-LO-phonon quantum kinetics and the generalized Kadanoff-Baym approximation,” Phys. Rev. B 60, 14234–14241 (1999).
[CrossRef]

G. Compagno and D. Valenti, “Long-time dynamics of self-dressing,” J. Phys. B 32, 4705–4717 (1999).
[CrossRef]

Q. T. Vu, L. Bányai, H. Haug, F. X. Camescasse, J.-P. Likforman, and A. Alexandrou, “Screened Coulomb quantum kinetics for resonant femtosecond spectroscopy in semiconductors,” Phys. Rev. B 59, 2760–2767 (1999).
[CrossRef]

1998

S. Tomimoto, H. Nansei, S. Saito, T. Suemoto, J. Takeda, and S. Kurita, “Femtosecond dynamics of the exciton self-trapping process in a quasi-one-dimensional halogen-bridged platinum complex,” Phys. Rev. Lett. 81, 417–420 (1998).
[CrossRef]

1997

Y. Zhao, D. W. Brown, and K. Lindenberg, “Variational energy band theory for polarons: mapping polaron structure with the Merrifield method,” J. Chem. Phys. 106, 5622–5630 (1997).
[CrossRef]

1995

R. Passante, T. Petrovsky, and G. Prigogine, “Long-time behaviour of self-dressing and indirect spectroscopy,” Physica A 218, 437–456 (1995).
[CrossRef]

1994

K. E. Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. B 50, 1541–1550 (1994).
[CrossRef]

1993

R. Passante, T. Petrovsky, and G. Prigogine, “Virtual transitions, self-dressing and indirect spectroscopy,” Opt. Commun. 99, 55–60 (1993).
[CrossRef]

G. Compagno, R. Passante, and F. Persico, “Partially dressed states and van Hove theory of quantum fields,” Il Nuovo Cimento D 15, 355–363 (1993).
[CrossRef]

1988

G. Compagno, R. Passante, and F. Persico, “Dressed and half dressed neutral sources in non relativistic QED,” Phys. Scr. T21, 33–39 (1988).
[CrossRef]

1987

D. W. Brown, K. Lindenberg, B. J. West, J. A. Cina, and R. Silbey, “Polaron formation in the acoustic chain,” J. Chem. Phys. 87, 6700–6705 (1987).
[CrossRef]

G. Iadonisi and F. Bassani, “Polaronic correction to the exciton effective mass,” Il Nuovo Cimento D 9, 703–714 (1987).
[CrossRef]

1986

D. W. Brown, K. Lindenberg, and B. J. West, “On the dynamics of polaron formation in a deformable medium,” J. Chem. Phys. 84, 1574–1582 (1986).
[CrossRef]

1985

G. Compagno, F. Persico, and R. Passante, “Interference in the virtual photon clouds of two hydrogen atoms,” Phys. Lett. A 112, 215–219 (1985).
[CrossRef]

1980

E. L. Feinberg, “Hadron clusters and half-dressed particles in quantum field theory,” Sov. Phys. Usp. 23, 629–650 (1980).
[CrossRef]

1964

R. E. Merrifield, “Theory of the vibrational structure of molecular exciton states,” J. Chem. Phys. 40, 445–450 (1964).
[CrossRef]

1959

T. Holstein, “Studies of polaron motion: Part II. The “small” polaron,” Ann. Phys. 8, 343–389 (1959).
[CrossRef]

T. Holstein, “Studies of polaron motion: Part I. The molecular-crystal model,” Ann. Phys. 8, 325–342 (1959).
[CrossRef]

1956

L. Van Hove, “Energy corrections and persistent perturbation effects in continuous spectra: II. The perturbed stationary states,” Physica 22, 343–354 (1956).
[CrossRef]

1955

L. Van Hove, “Energy corrections and persistent perturbation effects in continuous spectra,” Physica 21, 901–923 (1955).
[CrossRef]

G. C. Wick, “Introduction to some recent work in meson theory,” Rev. Mod. Phys. 27, 339–362 (1955).
[CrossRef]

1953

T. D. Lee, F. E. Low, and D. Pines, “The motion of slow electrons in a polar crystal,” Phys. Rev. 90, 297–302 (1953).
[CrossRef]

M. Gell-Mann and M. L. Goldberger, “The formal theory of scattering,” Phys. Rev. 91, 398–408 (1953).
[CrossRef]

1952

J. Pirenne, “Covariant theory of radiation damping,” Phys. Rev. 86, 395–398 (1952).
[CrossRef]

L. Van Hove, “Les difficultés de divergences pour un modèle particulier de champ quantifié,” Physica 18, 145–159 (1952).
[CrossRef]

1950

H. Frölich, H. Pelzer, and S. Zienau, “Properties of slow electrons in polar materials,” Philos. Mag. 41, 221–242 (1950).

1949

F. J. Dyson, “The S matrix in quantum electrodynamics,” Phys. Rev. 75, 1736–1755 (1949).
[CrossRef]

R. P. Feynman, “Space-time approach to quantum electrodynamics,” Phys. Rev. 76, 769–789 (1949).
[CrossRef]

1948

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372(1948).
[CrossRef]

J. Pirenne, “La methode des perturbations en theorie des champs quantifies et la construction de la matrice S de Heisenberg,” Helv. Phys. Acta 21, 226–232 (1948).

1947

W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243(1947).
[CrossRef]

H. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[CrossRef]

Abstreiter, G.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Alexandrou, A.

Q. T. Vu, L. Bányai, H. Haug, F. X. Camescasse, J.-P. Likforman, and A. Alexandrou, “Screened Coulomb quantum kinetics for resonant femtosecond spectroscopy in semiconductors,” Phys. Rev. B 59, 2760–2767 (1999).
[CrossRef]

An, Z.

Z. An, C. Q. Wu, and X. Sun, “Dynamics of photogenerated polarons in conjugated polymers,” Phys. Rev. Lett. 93, 216407(2004).
[CrossRef] [PubMed]

Bányai, L.

P. Gartner, L. Bányai, and H. Haug, “Self-consistent RPA for the intermediate-coupling polaron,” Phys. Rev. B 66, 075205(2002).
[CrossRef]

P. Gartner, L. Bányai, and H. Haug, “Two-time electron-LO-phonon quantum kinetics and the generalized Kadanoff-Baym approximation,” Phys. Rev. B 60, 14234–14241 (1999).
[CrossRef]

Q. T. Vu, L. Bányai, H. Haug, F. X. Camescasse, J.-P. Likforman, and A. Alexandrou, “Screened Coulomb quantum kinetics for resonant femtosecond spectroscopy in semiconductors,” Phys. Rev. B 59, 2760–2767 (1999).
[CrossRef]

K. E. Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. B 50, 1541–1550 (1994).
[CrossRef]

Bassani, F.

G. Iadonisi and F. Bassani, “Polaronic correction to the exciton effective mass,” Il Nuovo Cimento D 9, 703–714 (1987).
[CrossRef]

Bethe, H.

H. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[CrossRef]

Bichler, M.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Brodschelm, A.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Brown, C. T. A.

Brown, D. W.

Y. Zhao, D. W. Brown, and K. Lindenberg, “Variational energy band theory for polarons: mapping polaron structure with the Merrifield method,” J. Chem. Phys. 106, 5622–5630 (1997).
[CrossRef]

D. W. Brown, K. Lindenberg, B. J. West, J. A. Cina, and R. Silbey, “Polaron formation in the acoustic chain,” J. Chem. Phys. 87, 6700–6705 (1987).
[CrossRef]

D. W. Brown, K. Lindenberg, and B. J. West, “On the dynamics of polaron formation in a deformable medium,” J. Chem. Phys. 84, 1574–1582 (1986).
[CrossRef]

Brunner, F.

Burns, D.

Calvez, S.

Camescasse, F. X.

Q. T. Vu, L. Bányai, H. Haug, F. X. Camescasse, J.-P. Likforman, and A. Alexandrou, “Screened Coulomb quantum kinetics for resonant femtosecond spectroscopy in semiconductors,” Phys. Rev. B 59, 2760–2767 (1999).
[CrossRef]

Casimir, H. B. G.

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372(1948).
[CrossRef]

Cerullo, G.

D. Polli, L. Lüer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108–103116 (2007).
[CrossRef] [PubMed]

Chen, Q.-H.

Y.-Y. Zhang, Q.-H. Chen, and K.-L. Wang, “Quantum phase transition in the sub-Ohmic spin-boson model: an extended coherent-state approach,” Phys. Rev. B 81, 121105 (2010).
[CrossRef]

Cina, J. A.

D. W. Brown, K. Lindenberg, B. J. West, J. A. Cina, and R. Silbey, “Polaron formation in the acoustic chain,” J. Chem. Phys. 87, 6700–6705 (1987).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, 1992).

C. Cohen-Tannoudji, J. Dupont-Roc, and F. Laloë, Quantum Mechanics (Wiley, 1977).

Compagno, G.

G. Compagno and D. Valenti, “Radiative emission due to atomic self-dressing in QED,” Phys. Rev. A 65, 032106 (2002).
[CrossRef]

D. Valenti and G. Compagno, “Self-dressing dynamics of slow electrons in covalent crystals,” AIP Conf. Proc. 513, 242–245(2000).
[CrossRef]

G. Compagno and D. Valenti, “Long-time dynamics of self-dressing,” J. Phys. B 32, 4705–4717 (1999).
[CrossRef]

G. Compagno, R. Passante, and F. Persico, “Partially dressed states and van Hove theory of quantum fields,” Il Nuovo Cimento D 15, 355–363 (1993).
[CrossRef]

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[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Spectral distribution I ( ω , ω ) is shown for 0 < ω < ω D , 0 < ω < ω D (first Brillouin zone) using arbitrary unit (a.u.). The values of ω and ω are expressed in rad s 1 . The energy spectrum is almost everywhere flat and characterized by very small values. One peak is present in ω = 2 × 10 12 rad s 1 , ω = 7 × 10 12 rad s 1 and, symmetrically, another one in ω = 7 × 10 12 rad s 1 , ω = 2 × 10 12 rad s 1 . Moreover, around these two maxima the spectral distribution presents other smaller peaks corresponding to the emission of two phonons, one with frequency around 2 × 10 12 rad s 1 , the other one with frequency varying approximately between 6 × 10 12 rad s 1 and 10 13 rad s 1 .

Equations (46)

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H = H 0 + H I ,
H 0 = H e + H f = k E ( k ) a k a k + q ω ( q ) ( b q b q + 1 2 )
H I = 1 N k , q F l a ( q ) a k + q a k ( b q b q )
| ψ k 0 = | k ; { 0 q } = | k | { 0 q } ,
A e ( t ) = ψ k 0 | e i H t | ψ k 0 = 1 2 π i + G e ( E + i η ) e i E t d E .
G e ( E + i η ) = 1 E + i η E ( k ) R e ( E + i η ) ,
R e ( E + i η ) = ψ k 0 | H I | ψ k 0 + ψ k 0 | H I Q E + i η H 0 H I | ψ k 0 + ψ k 0 | H I Q E + i η H 0 H I Q E + i η H 0 H I | ψ k 0 +
R e ( E + i η ) = 4 9 E F 2 2 M N c l a q q E + i η E ( k q ) ω ( q ) .
E * ( k q ) = 2 ( k q ) 2 2 m ˜ Δ E ,
Δ E = π 72 α E F
R e ( E + i η ) = 4 9 E F 2 2 M N c l a q q E + i η E * ( k q ) ω ( q ) .
R e ( E + i η ) = Δ e + i Γ e 2
Δ e = 10 1 α E F , Γ e = 10 2 α 2 E F .
A e ( t ) = e i E ( k ) Δ e t e Γ e 2 t .
τ e = ( Γ e / ) 1 .
Δ e = 10 2 eV , Γ e = 10 5 eV .
τ e 10 11 s .
H | ψ ˜ e = E ˜ e | ψ ˜ e .
| ψ ˜ e = Z | ψ k 0 + Q k | ψ ˜ e .
Q k | ψ ˜ e = Z Q k E * ( k ) + i η Q k H Q k H I P k | ψ k 0 = Z ( Q k E * ( k ) + i η Q k H 0 Q k H I P k + Q k E * ( k ) + i η Q k H 0 Q k H I Q k E * ( k ) + i η Q k H 0 Q k H I P k + ) | ψ k 0 .
| ψ ˜ e = Z [ | ψ k 0 + ( Q k E * ( k ) + i η Q k H 0 Q k H I P k + Q k E * ( k ) + i η Q k H 0 Q k H I Q k E * ( k ) + i η Q k H 0 Q k H I P k + ) | ψ k 0 ] .
ψ ˜ e ; { n q } | U ( t , 0 ) | ψ k 0 = 1 2 π i C d z ψ ˜ e ; { n q } | G ( z ) | ψ k 0 e i z t ,
ψ ˜ e ; q | U ( t , 0 ) | ψ k 0 = 1 2 π i C d z ψ ˜ e ; q | G ( z ) | ψ k 0 e i z t .
b q G ( z ) = G ( z ω ) b q + G ( z ω ) V q G ( z )
V q = [ b q , H I ] = 1 N k F l a ( q ) a k q a k .
ψ ˜ e ; q | G ( z ) | ψ k 0 = ψ ˜ e | G ( z ω ) V q G ( z ) | ψ k 0 .
ψ ˜ e ; q | U ( t , 0 ) | ψ k 0 = 0 .
ψ ˜ e | b q b q G ( z ) | ψ k 0 = ψ ˜ e | { G ( z ω ω ) b q + G ( z ω ω ) V q G ( z ω ) } b q | ψ k 0 + ψ ˜ e | { G ( z ω ω ) b q + G ( z ω ω ) V q G ( z ω ) } V q G ( z ) | ψ k 0 .
ψ ˜ e | b q b q G ( z ) | ψ k 0 = ψ ˜ e | G ( z ω ω ) V q { G ( z ω ) b q + G ( z ω ) V q G ( z ) } | ψ k 0 + ψ ˜ e | G ( z ω ω ) V q G ( z ω ) V q G ( z ) | ψ k 0 .
ψ ˜ e | b q b q G ( z ) | ψ k 0 = ψ ˜ e | G ( z ω ω ) V q G ( z ω ) V q G ( z ) | ψ k 0 + ψ ˜ e | G ( z ω ω ) V q G ( z ω ) V q G ( z ) | ψ k 0 .
ψ ˜ e | G ( z ħ ω ħ ω ) V q G ( z ħ ω ) V q G ( z ) | ψ k 0 = 1 z E * ( k ) ħ ω ħ ω ψ ˜ e | V q G ( z ħ ω ) V q G ( z ) | ψ k 0 .
ψ ˜ e | V q G ( z ħ ω ) V q G ( z ) | ψ k 0 = F l a ( q ) F l a ( q ) 1 N k k ψ ˜ e | a k q a k G ( z ħ ω ) a k q × a k G ( z ) | ψ k 0 .
a k q a k = | ψ k q 0 ψ k 0 | .
ψ ˜ e | V q G ( z ω ) V q G ( z ) | ψ k 0 = F l a ( q ) F l a ( q ) 1 N k k ψ e | ψ k q 0 ψ k 0 | G ( z ω ) | ψ k q 0 ψ k 0 | G ( z ) | ψ k 0 .
ψ k 0 | G ( z ) | ψ k 0 = ψ k 0 | G ( z ) | ψ k 0 δ k , k ,
ψ k 0 | G ( z ω ) | ψ k q = ψ k 0 | G ( z ω ) | ψ k 0 δ k , k q .
ψ ˜ e | V q G ( z ω ) V q G ( z ) | ψ k 0 = F l a ( q ) F l a ( q ) 1 N ψ e | ψ k q q 0 ψ k q 0 | G ( z ω ) | ψ k q 0 ψ k 0 | G ( z ) | ψ k 0 .
ψ ˜ e | V q G ( z ω ) V q G ( z ) | ψ k 0 = F l a ( q ) F l a ( q ) 1 N Z ψ k q 0 | G ( z ω ) | ψ k q 0 ψ k 0 | G ( z ) | ψ k 0 .
ψ ˜ e | G ( z ω ω ) V q G ( z ω ) V q G ( z ) | ψ k 0 = 1 N Z F l a ( q ) F l a ( q ) × 1 z E * ( k ) ω ω ψ k q 0 | G ( z ω ) | ψ k q 0 ψ k 0 | G ( z ) | ψ k 0 .
ψ k 0 | G ( z ) | ψ k 0 = 1 z E * ( k ) i 2 Γ e ,
ψ k q 0 | G ( z ω ) | ψ k q 0 = 1 z E * ( k q ) ω i 2 Γ e ,
ψ e | G ( z ω ω ) V q G ( z ω ) V q G ( z ) | ψ k 0 = 1 N Z F l a ( q ) F l a ( q ) × 1 z E * ( k ) ω ω × 1 z E * ( k q ) ω i 2 Γ e × 1 z E * ( k ) i 2 Γ e .
ψ e | b q b q G ( z ) | ψ k 0 = 1 N Z F l a ( q ) F l a ( q ) × { 1 z E * ( k ) ω ω × 1 z E * ( k q ) ω i 2 Γ e × 1 z E * ( k ) i 2 Γ e + 1 z E * ( k ) ω ω × 1 z E * ( k q ) ω i 2 Γ e × 1 z E * ( k ) i 2 Γ e } .
ψ e ; q , q | U ( t , 0 ) | ψ k 0 = 1 N Z F l a ( q ) F l a ( q ) exp [ i ( E * ( k ) + ω + ω ) t ] × { 1 ω 2 2 m [ q 2 2 k q sin θ ] + i 2 Γ e × 1 ω + ω + i 2 Γ e + 1 ω 2 2 m [ q 2 2 k q sin θ ] + i 2 Γ e × 1 ω + ω + i 2 Γ e } ,
L 3 2 π ω 2 d ω ( 2 π ) 3 L 3 2 π ω 2 d ω ( 2 π ) 3 ,
I ( ω , ω ) = L 3 4 π ( 2 π ) 3 L 3 4 π ( 2 π ) 3 1 N 2 Z 4 9 E F 2 2 M c l a 2 4 9 E F 2 2 M c l a 2 1 2 × ω 3 ω 3 { 1 ( ω ω 2 2 m c l a 2 ) 2 + Γ 2 4 2 × 1 ( ω + ω ) 2 + Γ 2 4 2 + 1 ( ω ω 2 2 m c l a 2 ) 2 + Γ 2 4 2 × 1 ( ω + ω ) 2 + Γ 2 4 2 } .

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