Abstract

Entangled photons are the fundamental resource in quantum information processing. How to produce them efficiently has always been a matter of concern. Here we propose a new way to produce correlated photons efficiently from monolayer WS2 based on bound states in the continuum (BICs). The BICs of radiation modes in the monolayer WS2 are realized by designing the photonic crystal slab-WS2-slab structure. The generation efficiency of correlated photon pairs from such a structure has been studied by using a rigorous quantum model of spontaneous parametric down-conversion with the plane wave expansion method. It is found that the generation efficiency of correlated photon pairs is greatly improved if the signal and idler fields are located at the BICs determined by the inverse scattering matrix of the structure. This is in contrast to the parametric down-conversion process for the enhanced generation of nonlinear waves if the pump field is located at the BICs determined by the scattering matrix of the structure. The generation rate of the correlated photon pairs can be improved by 7 orders of magnitude in some designed structures. The generated quantum signals are sensitive to the wavelength and exhibit narrowed relative line width, which is very beneficial for quantum information processing.

© 2019 Chinese Laser Press

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Improved third-order nonlinear effect in graphene based on bound states in the continuum

Tiecheng Wang and Xiangdong Zhang
Photon. Res. 5(6) 629-639 (2017)

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2018 (2)

K. Niizeki, K. Ikeda, M. D. Zheng, X. P. Xie, K. Okamura, N. Takei, N. Namekata, S. Inoue, H. Kosaka, and T. Y. Horikiri, “Ultrabright narrow-band telecom two-photon source for long-distance quantum communication,” Appl. Phys. Express 11, 042801 (2018).
[Crossref]

D. Huber, M. Reindl, J. Aberl, A. Rastelli, and R. Trotta, “Semiconductor quantum dots as an ideal source of polarization entangled photon pairs on-demand: a review,” J. Opt. 20, 073002 (2018).
[Crossref]

2017 (4)

L. Wang, T. Wang, S. Zhang, P. Xie, and X. D. Zhang, “Larger enhancement in four-wave mixing from graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 34, 2000–2010 (2017).
[Crossref]

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196–199 (2017).
[Crossref]

T. Wang and X. D. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5, 629–639 (2017).
[Crossref]

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Beams 20, 024701 (2017).
[Crossref]

2016 (5)

E. N. Bulgakov and A. F. Sadreev, “Transfer of spin angular momentum of an incident wave into orbital angular momentum of the bound states in the continuum in an array of dielectric spheres,” Phys. Rev. A 94, 033856 (2016).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second-and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94, 035435 (2016).
[Crossref]

S. Zhang and X. D. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33, 452–460 (2016).
[Crossref]

J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33, 615–621 (2016).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third-harmonic generation in periodically patterned graphene and transition monolayers,” Phys. Rev. B 94, 035435 (2016).
[Crossref]

2015 (3)

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a WSe2 monolayer transistor,” Nat. Nanotechnol. 10, 407–411 (2015).
[Crossref]

M. Zhang and X. D. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5, 8266 (2015).
[Crossref]

E. N. Bulgakov, K. N. Pichugin, and A. F. Sadreev, “All-optical light storage in bound states in the continuum and release by demand,” Opt. Express 23, 22520–22531 (2015).
[Crossref]

2014 (7)

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90, 205422 (2014).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical perspective for bound states in the continuum in photonic crystal slabs,” Phys. Rev. Lett. 113, 037401 (2014).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Nonlinear processes in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89, 165139 (2014).
[Crossref]

F. Monticone and A. Alù, “Embedded photonic eigenvalues in 3D nanostructures,” Phys. Rev. Lett. 112, 213903 (2014).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 89, 023813 (2014).
[Crossref]

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary second harmonic generation in tungsten disulfide monolayers,” Sci. Rep. 4, 5530 (2014).
[Crossref]

2013 (5)

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188–191 (2013).
[Crossref]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2, e84 (2013).
[Crossref]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation,” Nano Lett. 13, 3329–3333 (2013).
[Crossref]

T. R. Zhan, X. Shi, Y. Y. Dai, X. H. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25, 215301 (2013).
[Crossref]

2012 (2)

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108, 070401 (2012).
[Crossref]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref]

2011 (1)

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental observation of optical bound states in the continuum,” Phys. Rev. Lett. 107, 183901 (2011).
[Crossref]

2010 (1)

S. Wei, Y. Dong, H. Wang, and X. D. Zhang, “Enhancement of correlated photon-pair generation from a positive-negative index material heterostructure,” Phy. Rev. A 81, 053830 (2010).
[Crossref]

2008 (3)

X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[Crossref]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref]

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78, 075105 (2008).
[Crossref]

2007 (2)

Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Agren, “Highly efficient generation of entangled photon pair by spontaneous parametric downconversion in defective photonic crystals,” J. Opt. Soc. Am. B 24, 1365–1368 (2007).
[Crossref]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[Crossref]

2006 (1)

L. Sciscione, M. Centini, C. Sibilia, M. Bertolotti, and M. Scalora, “Entangled, guided photon generation in (1+1)-dimensional photonic crystals,” Phys. Rev. A 74, 013815 (2006).
[Crossref]

2005 (5)

W. T. M. Irvine, M. J. A. de Dood, and D. Bouwmeester, “Bloch theory of entangled photon generation in non-linear photonic crystals,” Phys. Rev. A 72, 043815 (2005).
[Crossref]

M. Centini, J. Perina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, “Entangled photon pair generation by spontaneous parametric down-conversion in finite-length one-dimensional photonic crystals,” Phys. Rev. A 72, 033806 (2005).
[Crossref]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[Crossref]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534–544 (2005).
[Crossref]

J. Fulconis, O. Alibart, W. J. Wadsworth, P. S. Russell, and J. G. Rarity, “High brightness single mode source of correlated photon pairs using a photonic crystal fiber,” Opt. Express 13, 7572–7582 (2005).
[Crossref]

2004 (2)

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum ellipsometry using correlated-photon beams,” Phys. Rev. A 70, 043810 (2004).
[Crossref]

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93, 040504 (2004).
[Crossref]

2003 (1)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-bases transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[Crossref]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

1998 (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[Crossref]

1996 (1)

T. Gruner and D.-G. Welsh, “Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous Kramers-Kronig dielectrics,” Phys. Rev. A 53, 1818–1829 (1996).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref]

Aberl, J.

D. Huber, M. Reindl, J. Aberl, A. Rastelli, and R. Trotta, “Semiconductor quantum dots as an ideal source of polarization entangled photon pairs on-demand: a review,” J. Opt. 20, 073002 (2018).
[Crossref]

Agren, H.

Akozbek, N.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[Crossref]

Alencar, T. V.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
[Crossref]

Alibart, O.

Alù, A.

F. Monticone and A. Alù, “Embedded photonic eigenvalues in 3D nanostructures,” Phys. Rev. Lett. 112, 213903 (2014).
[Crossref]

Aryshev, A.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Beams 20, 024701 (2017).
[Crossref]

Bahari, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196–199 (2017).
[Crossref]

Bao, X. H.

X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[Crossref]

Barboza, A. P. M.

L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87, 201401 (2013).
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Figures (7)

Fig. 1.
Fig. 1. (a) Diagram of the photonic crystal slab-monolayer WS2-slab. The air holes are arranged in a square lattice with lattice constant l and the radius of the holes is r. The thicknesses of the photonic crystal slab and dielectric slab are denoted by d1 and d2, and the monolayer WS2 is put at the interface between the photonic crystal slab and the dielectric slab. (b) Schematic of the photon-pair generation process in the three-layer structure. The pump beam with frequency ωp and angle θp is incident on the three-layer structure, and due to the second-order nonlinear effect of the monolayer WS2, the signal field with frequency ωs and angle θs and the idler field with frequency ωi and angle θi are generated.
Fig. 2.
Fig. 2. (a), (b), and (c) show the transmission (t), reflection (r), and absorption (a) spectra for the three-layer structure as shown in Fig. 1, respectively. (d), (e) and (f) display the corresponding transmission (t), reflection (r), and absorption (a) spectra, respectively.
Fig. 3.
Fig. 3. (a) and (b) describe the downward (black line) and upward (red line) SHG conversion from the three-layer structure as a function of the wavelength of the pump field in different regions, and the corresponding SHG conversions from the freestanding monolayer WS2 are shown by the corresponding dashed lines. (c) and (d) exhibit energy SsFF (black line), SsFB (red line), SsBF (green line), and SsBB (blue line) of the three-layer structure as a function of the wavelength of the signal field in different regions, the corresponding energy spectra of the freestanding monolayer WS2 are represented by the dashed lines, and the wavelength of the pump field is λp=749.627  nm. The parameters are assumed as follows: d1=1.00l, d2=1.70l, r=0.20l, and l=700  nm.
Fig. 4.
Fig. 4. (a) and (b) exhibit the electric field enhancements |EW| (black line) and |EW| (red line) determined by the scattering matrix and inverse scattering matrix of the three-layer structure as a function of the wavelength. (c) and (d) describe the absorption a of the three-layer structure as a function of the wavelength in different regions; the corresponding absorption a of the freestanding monolayer WS2 is shown by the corresponding red lines. (e) and (f) show the absorption a of the three-layer structure as a function of the wavelength in different regions; the corresponding absorption a of the freestanding monolayer WS2 is represented by the red lines.
Fig. 5.
Fig. 5. (a) The absorption a of the three-layer structure as a function of the wavelength λ at various incident angles; the black, red, and green lines are the spectra at the incident angles θ=5°, 10°, and 15°, respectively. (b) The energy spectra SsFF of the three-layer structure as a function of the wavelength λs at various radiated angles of the signal field θs=5°, 10°, and 15°; the corresponding energy spectra of the freestanding monolayer WS2 are denoted by the dashed lines. The wavelengths of the pump fields which are incident normally are taken as λp=774.808  nm, 797.856 nm, and 820.341 nm for various θs, respectively; the other parameters are assumed as follows: d1=1.00l, d2=1.70l, r=0.20l, and l=700  nm.
Fig. 6.
Fig. 6. (a) The absorption a of the three-layer structure as a function of the incident angle θ. (b) The energy spectra SsFF (black line), SsFB (red line with circle), SsBF (green line with triangle), and SsBB (blue line) of the three-layer structure as a function of the radiated angle θs of the signal field. The wavelength of the pump field, which is incident normally, is taken as λp=749.627  nm, and the wavelengths of the signal and idler fields are fixed at λs=λi=2λp=1499.254  nm, the other parameters are assumed as follows: d1=1.00l, d2=1.70l, r=0.20l, and l=700  nm.
Fig. 7.
Fig. 7. (a), (b), and (c) The energy SsFF as a function of the wavelength of the signal field under normal incident pump field, the signal and idler fields are also radiated normally, and the corresponding spectra of the freestanding monolayer WS2 are described by the dashed lines. (a) Various radii of the air hole; the wavelengths of the pump fields are fixed at λp=752.828  nm, 751.173 nm, 749.627 nm, and 748.271 nm when the radii are r=0.01l, 0.15l, 0.20l, and 0.25l. The other parameters are taken as follows: d1=1.00l and d2=1.70l. (b) Various thicknesses of the photonic crystal slab; the wavelengths of the pump fields are fixed at λp=749.551  nm, 749.627 nm, 749.694 nm, and 749.749 nm when the thicknesses are d1=0.95l, 1.00l, 1.05l, and 1.10l. The other parameters are taken as follows: d2=1.70l and r=0.20l. (c) Various thicknesses of the dielectric slab; the wavelengths of the pump fields are fixed at λp=744.405  nm, 749.627 nm, 754.555 nm, and 759.204 nm when the thicknesses are d2=1.65l, 1.70l, 1.75l, and 1.80l. The other parameters are taken as follows: d1=1.00l and r=0.20l. (d) The energy SsFF for different polarizations of generated photons as a function of the azimuthal angle of the normal incident pump field; the signal and idler fields are also radiated normally and are polarized in the yy, yx, xy, xx directions. The parameters are assumed as follows: d1=1.00l, d2=1.70l, r=0.20l, and λs=λi=2λp=1499.254  nm.

Equations (48)

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χ(2)=χyyy(2)=χyxx(2)=χxyx(2)=χxxy(2),
Hint(t)=ε0drα,β,γ[χαβγ(2)Ep,α+(r,t)E^s,β(r,t)E^i,γ(r,t)+H.c.],
Ep,α+(r,t)=mn[EpFmn,αexp(iβpmnz)+EpBmn,αexp(iβpmnz)]×exp[i(kpmnxx+kpmnyy)iωpt]=mnEpmn,α+(z,ωp)exp[i(kpmnxx+kpmnyy)iωpt],
βpmn=kp2kpmnx2kpmny2,kp=ωpn(ωp)/c,
E^v,α(r,t)=0dωvmnGvmn[a^vFmn,α2exp(iβvmnz)+a^vBmn,α2exp(iβvmnz)]exp[i(kvmnxx+kvmnyy)iωvt]=0dωvmnE^vmn,α(z,ων)exp[i(kvmnxx+kvmnyy)iωvt],
Gvmn=ε0ω2εI(ωv)8πβvmnIβvmn2,
βvmn=kv2kvmnx2kvmny2,kv=ωvcn(ωv).
(a^vF4a^vB0)=(QIQIIQIIIQIV)(a^vF0a^vB4)=Q(a^vF0a^vB4),
(a^vF2a^vB2)=(TI1,2TII1,2TIII1,2TIV1,2)(a^vF0a^vB0)=T1,2(a^vF0a^vB0),
(a^vF2a^vB2)=(TITIITIIITIV)((QIIIQIVQII1QI)1QIVQII1(QIIIQIVQII1QI)101)(a^vF4a^vB0)=(Fv11Fv12Fv21Fv22)(a^vF4a^vB0).
Q1*=(QI1*QII1*QIII1*QIV1*)=(QIQIIQIIIQIV)1*=((QIIIQIVQII1QI)1QIVQII1(QIIIQIVQII1QI)1QII1+QII1QI(QIIIQIVQII1QI)1QIVQII1QII1QI(QIIIQIVQII1QI)1)*.
EFmn,αtr=pqβQI,mnα,pqβEFpq,βin,
EBmn,αrf=pqβQIII,mnα,pqβEFpq,βin.
Itr(2ω)=12ε0α,mn[E3,α,mn+(2ω)][E3,α,mn+(2ω)]*,
Irf(2ω)=12ε0α,mn[E0,α,mn(2ω)][E0,α,mn(2ω)]*,
Iin(ω)=12ε0α,mn[Ein,α,mn+(ω)][Ein,α,mn+(ω)]*.
F=Itr(2ω)+Irf(2ω)Iin(ω).
t=mnαEFmn,αtrEFmn,αtr*βmnpqαEFpq,αinEFpq,αin*βpq,
r=mnαEBmn,αrfEBmn,αrf*βmnpqαEFpq,αinEFpq,αin*βpq.
a=1tr.
Hint(t)=ε0B2π0dωs0dωidzχαβγ(2)mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)Epmn,α+(z,ωp)E^sot,β(z,ωs)E^irs,γ(z,ωi)×exp[i(ωpωsωi)]+H.c..
kpmnxksotxkirsx=0,
kpmnyksotykirsy=0.
|ψs,β,i,γout=exp[idtHint(t)]|vac=|vacidtHint(t)|vac.
|ψs,β,i,γout=|vaciε0B2π2π0dωs0dωiχ(2):mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)×w=pF,pBg=sF,sBh=iF,iBdwGsot*Girs*Ewmna^got2+a^hrs2+×exp[(βwmnβgot*βhrs*)dw/2]×sinc[(βwmnβgot*βhrs*)d/2]δ(ωpωsωi)|vac.
ωp=ωs+ωi.
|ψs,β,i,γ(2)=|ψs,β,i,γFF+|ψs,β,i,γFB+|ψs,β,i,γBF+|ψs,β,i,γBB,
|ψs,β,i,γFF=0dωs0dωi[ϕFF(ωs,ωi)a^sF00,β4+(ωs)×a^iF00,γ4+(ωi)δ(ωpωsωi)]|vac,
|ψs,β,i,γFB=0dωs0dωi[ϕFB(ωs,ωi)a^sF00,β4+(ωs)×a^iF00,γ0+(ωi)δ(ωpωsωi)]|vac,
|ψs,β,i,γBF=0dωs0dωi[ϕBF(ωs,ωi)a^sF00,β0+(ωs)×a^iF00,γ4+(ωi)δ(ωpωsωi)]|vac,
|ψs,β,i,γBB=0dωs0dωi[ϕBB(ωs,ωi)a^sF00,β0+(ωs)×a^iF00,γ0+(ωi)δ(ωpωsωi)]|vac,
ϕFF(ωs,ωi)=iε0B2π2πχ(2):mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)exp[(βwmnβgot*βhrs*)dw/2]×w=pFw=pBg=sF(b=11)g=sB(b=21)h=iF(c=11)h=iB(c=21)dw×Gsot*Girs*Ewmn(Fsb)ot,00*(Fic)rs,00*×sinc[(βwmnβgot*βhrs*)dw/2],
ϕFB(ωs,ωi)=iε0B2π2πχ(2):mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)exp[(βwmnβgot*βhrs*)dw/2]×w=pFw=pBg=sF(b=11)g=sB(b=21)h=iF(c=12)h=iB(c=22)dw×Gsot*Girs*Ewmn(Fsb)ot,00*(Fic)rs,00*×sinc[(βwmnβgot*βhrs*)dw/2],
ϕBF(ωs,ωi)=iε0B2π2πχ(2):mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)exp[(βwmnβgot*βhrs*)dw/2]×w=pFw=pBg=sF(b=12)g=sB(b=22)h=iF(c=11)h=iB(c=21)dw×Gsot*Girs*Ewmn(Fsb)ot,00*(Fic)rs,00*×sinc[(βwmnβgot*βhrs*)dw/2],
ϕBB(ωs,ωi)=iε0B2π2πχ(2):mn,ot,rsδ(kpmnxksotxkirsx)δ(kpmnyksotykirsy)exp[(βwmnβgot*βhrs*)dw/2]×w=pFw=pBg=sF(b=12)g=sB(b=22)h=iF(c=12)h=iB(c=22)dw×Gsot*Girs*Ewmn(Fsb)ot,00*(Fic)rs,00*,×sinc[(βwmnβgot*βhrs*)dw/2],
|ϕhk(ωs,ωi)|2=f(ωs,ωi)δ2(ωpωsωi)=limT2T2πf(ωs,ωi)δ(ωpωsωi),
f(ωs,ωi)=(2π)3/2(ε0Bc)2|ϕ(ωs,ωi)|2
ϕ(ωs,ωi)=mn,ot,rsχ(2):exp[(βwmnβgot*βhrs*)dw/2]×w=pFw=pBg=sF(b=hp)g=sB(b=hq)h=iF(c=kp)h=iB(c=kq)dw×Gsot*Girs*Ewmn(Fsb)ot,00*(Fic)rs,00*(Fp=11,Fq=21,Bp=12,Bq=22),
Ns,ihk(ωs,ωi)=ψs,β,i,γhk|n^sh,β(ωs)n^ik,γ(ωi)|ψs,β,i,γhk,
n^sh,α(ωs)=a^sh00,αa^sh00,α,
n^ik,α(ωi)=a^ik00,αa^ik00,α,
a^gF00,α=a^gF00,α4,
a^gB00,α=a^gB00,α0,g=s,i.
Ns,ihk(ωs,ωi)=|ϕhk(ωs,ωi)|2.
Nshk(ωs)=0dωi|ϕhk(ωs,ωi)|2.
Ssh,k(ωs)=ωsNsh,k(ωs)=ωs0dωi|ϕhk(ωs,ωi)|2.
Nsh,k(ωs)=12πf[ωs,(ωpωs)],
Ssh,k(ωs)=ωs2πf[ωs,(ωpωs)].