Abstract

Integrated quantum photonics relies critically on the purity, scalability, integrability, and flexibility of a photon source to support diverse quantum functionalities on a single chip. Here we report a chip-scale photon-pair source on the silicon-on-insulator platform that utilizes dramatic cavity-enhanced four-wave mixing in a high-Q silicon microdisk resonator. The device is able to produce high-quality photon pairs at different wavelengths with a high spectral brightness of 6.24×107 pairs/s/mW2/GHz and photon-pair correlation with a coincidence-to-accidental ratio of 1386 ± 278 while pumped with a continuous-wave laser. The superior performance, together with the structural compactness and CMOS compatibility, opens up a great avenue towards quantum silicon photonics with capability of multi-channel parallel information processing for both integrated quantum computing and long-haul quantum communication.

© 2015 Optical Society of America

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2015 (3)

2014 (3)

2013 (7)

J. B. Spring, P. S. Salter, B. J. Metcalf, P. C. Humphreys, M. Moore, N. Thomas-Peter, M. Barbieri, Xian-Min Jin, N. K. Langford, W. S. Kolthammer, M. J. Booth, and I. A. Walmsley, “On-chip low loss heralded source of pure single photons,” Opt. Express 21, 13522 (2013).
[Crossref] [PubMed]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, Xian-Min Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multi-photon quantum interference in a multi-port integrated photonic device,” Nature Commun. 4, 1356 (2013).
[Crossref]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in silicon microring resonators with reverse bias enhancement,” Opt. Express 21, 27826 (2013).
[Crossref]

N. Matsuda, H. Takesue, K. Shimizu, Y. Tokura, E. Kuramochi, and M. Notomi, “Slow light enhanced correlated photon pair generation in photonic-crystal coupled-resonator optical waveguides,” Opt. Express 21, 8596 (2013).
[Crossref] [PubMed]

R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 μm from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969 (2013).
[Crossref] [PubMed]

M. J. Collins, C. Xiong, I. H. Rey, T. D. Vo, J. He, S. Shahnia, C. Reardon, T. F. Krauss, M. J. Steel, A. S. Clark, and B. J. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nature Commun. 4, 2582 (2013).
[Crossref]

M. Förtsch, J. U. Fürst, C. Wittmann, D. Strekalov, A. Aiello, M. V. Chekhova, C. Silberhorn, G. Leuchs, and C. Marquardt, “A versatile source of single photons for quantum information processing,” Nature Commun. 4, 1818 (2013).
[Crossref]

2012 (5)

M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 100, 261104 (2012).
[Crossref]

S. Azzini, D. Grassani, M. J. Strain, M. Sorel, L. G. Helt, J. E. Sipe, M. Liscidini, M. Galli, and D. Bajoni, “Ultra-low power generation of twin photons in a compact silicon ring resonator,” Opt. Express 20, 23100 (2012).
[Crossref] [PubMed]

C.-S. Chuu, G. Y. Yin, and S. E. Harris, “A miniature ultrabright source of temporally long, narrowband biphotons,” Appl. Phys. Lett. 101, 051108 (2012).
[Crossref]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys. Rev. Lett. 108, 153605 (2012).
[Crossref] [PubMed]

2011 (7)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon sources and detectors,” Rev. Sci. Inst. 82, 071101 (2011).
[Crossref]

A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bongioanni, F. Sciarrino, G. Vallone, and P. Mataloni, “Integrated photonic quantum gates for polarization qubits,” Nat. Commun. 2, 566 (2011).
[Crossref] [PubMed]

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nature Photon. 5, 222 (2011).
[Crossref]

H. Zhang, Xian-Min Jin, J. Yang, Han-Ning Dai, Sheng-Jun Yang, Tian-Ming Zhao, J. Rui, Y. He, X. Jiang, F. Yang, Ge-Sheng Pan, Zhen-Sheng Yuan, Y. Deng, Zeng-Bing Chen, Xiao-Hui Bao, Shuai Chen, B. Zhao, and Jian-Wei Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nature Photon. 5, 628 (2011).
[Crossref]

C. Xiong, C. Monat, A. S. Clark, C. Grillet, G. D. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, J. G. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett. 36, 3413 (2011).
[Crossref] [PubMed]

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

2010 (5)

K. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S. Itabashi, “Frequency and Polarization Characteristics of Correlated Photon-Pair Generation Using a Silicon Wire Waveguide,” IEEE J. Sel. Top. Quantum Electron. 16, 325 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535 (2010).
[Crossref]

D. Liang and J. E. Bowers, “Recent progess in lasers on silicon,” Nature Photon. 4, 511 (2010).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nature Photon. 4, 518 (2010).
[Crossref]

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Phys. Rev. Lett. 105, 253601 (2010).
[Crossref]

2009 (6)

S. D. Dyer, B. Baek, and S. W. Nam, “High-brightness, low-noise, all-fiber photon pair source,” Opt. Express 17, 10290 (2009).
[Crossref] [PubMed]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
[Crossref] [PubMed]

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558 (2009).
[Crossref] [PubMed]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon. 3, 687 (2009).
[Crossref]

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17, 6727 (2009).
[Crossref] [PubMed]

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102, 063603 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (8)

N. Gisin and R. Thew, “Quantum communication,” Nature Photon. 1, 165 (2007).
[Crossref]

P. Kok, W. J. Munro, K. Nemoto, and T. C. Ralph, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79135 (2007).
[Crossref]

Q. Zhang, X. Xie, H. Takesue, S. W. Nam, C. Langrock, M. M. Fejer, and Y. Yamamoto, “Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock,” Opt. Express 15, 10288 (2007).
[Crossref] [PubMed]

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15, 15377 (2007).
[Crossref] [PubMed]

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007).
[Crossref]

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A 75, 023803 (2007).
[Crossref]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604 (2007).
[Crossref] [PubMed]

J. Fan and A. Migdall, “A broadband high spectral brightness fiber-based two-photon source,” Opt. Express 15, 2915 (2007).
[Crossref] [PubMed]

2006 (3)

O. Alibart, J. Fulconis, G. K. L. Wong, S. G. Murdoch, W. J. Wadsworth, and J. G. Rarity, “Photon pair generating using four-wave mixing in a microstructured fibre: theory versus experiment,” N. J. Phys. 8, 15606 (2006).
[Crossref]

Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. 31, 3140 (2006).
[Crossref] [PubMed]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Exp. 14, 12388 (2006).
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Optica (1)

Phys. Rev. A (4)

M. G. Raymer, J. Noh, K. Banaszek, and I. A. Walmsley, “Pure-state single-photon wave-packet generation by parametric down-conversion in a distributed microcavity,” Phys. Rev. A 72, 023825 (2005).
[Crossref]

Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A 75, 023803 (2007).
[Crossref]

F. König, E. J. Mason, F. N. C. Wong, and M. A. Albota, “Efficient and spectrally bright source of polarizationentangled photons,” Phys. Rev. A 71, 033805 (2005).
[Crossref]

H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bells inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A 70, 031802(R) (2004).
[Crossref]

Phys. Rev. B (1)

P. A. Temple and C. E. Hathaway, “Multiphonon Raman spectrum of silicon,” Phys. Rev. B 7, 3685 (1973).
[Crossref]

Phys. Rev. Lett. (8)

M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric down-conversion,” Phys. Rev. Lett. 102, 063603 (2009).
[Crossref] [PubMed]

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240 (1994).
[Crossref] [PubMed]

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

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, “Bright source of spectrally uncorrelated polarization-entangled photons with nearly single-mode emission,” Phys. Rev. Lett. 105, 253601 (2010).
[Crossref]

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys. Rev. Lett. 108, 153605 (2012).
[Crossref] [PubMed]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
[Crossref] [PubMed]

Prog. Opt. (1)

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227 (2011).
[Crossref]

Rev. Mod. Phys. (2)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

P. Kok, W. J. Munro, K. Nemoto, and T. C. Ralph, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79135 (2007).
[Crossref]

Rev. Sci. Inst. (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon sources and detectors,” Rev. Sci. Inst. 82, 071101 (2011).
[Crossref]

Science (1)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646 (2008).
[Crossref] [PubMed]

Other (2)

D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2008).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1

a, Schematic of generating photon pairs from a silicon microdisk resonator sitting on a silica pedestal. All photon modes couple to the same propagation mode inside the delivery coupling waveguide, which is a single-mode tapered silica fiber. b, Group-velocity dispersion of vertically polarized transverse-magnetic-like (TM-like) cavity modes with four different radial orders, termed as P0, P1, P2, and P3, respectively, for a silicon microdisk with a thickness of 260 nm and a radius of 4.9 μm, simulated by the finite element method. The insets show the simulated optical field profiles.

Fig. 2
Fig. 2

Schematic of the experimental setup for optical spectroscopy and coincidence photon counting. The inset shows the scanning electron microscopic image of the fabricated silicon microdisk with a thickness of 260 nm and a radius of 4.9 μm.

Fig. 3
Fig. 3

The transmission spectrum of the passive cavity scanned by two tunable lasers operating at different spectral ranges (indicated as blue and green). The two sets of arrows indicate cavity modes with constant mode spacings within the P1 and P2 mode families, respectively. The insets show detailed transmission spectra of a P2 mode at 1497.2 nm and a P1 mode at 1564.9 nm, with theoretical fitting shown in red. See Appendix A for the discussion of cavity mode splitting.

Fig. 4
Fig. 4

a and b, The photoluminescence spectra of the P2 and P1 photon-pair comb, respectively, with the pump nearly critically coupled to the cavity. The colors indicate the individual spectra recorded at different transmission ports of the CWDM DEMUX. The four mode pairs in the two combs are termed as P 2 I , P 2 II , P 1 I , and P 1 II , respectively, as indicated on the figures.

Fig. 5
Fig. 5

Recorded photon emission fluxes, pair emission fluxes, and CARs for different photon-pair modes ( P 1 I , P 1 II , P 2 I , and P 2 II ). a–d, Recorded emission fluxes of individual photon modes within each mode pair. The signal (idler) at the shorter (longer) wavelength is shown as blue (red). Solid curves show the theoretical prediction of pure FWM-created photon fluxes. e–h, Recorded emission fluxes of correlated photon pairs. In each figure, the left and right vertical axes show the emission flux and the corresponding true coincidence count per gate, respectively, indicated by brown and green colors. The solid curves show the theoretical prediction. i–l, Recorded CAR for each mode pair. No accidental coincidence is subtracted. The red solid curves show the theoretical prediction. The black dashed curves are calculated from independently recorded true coincidence counts and photon fluxes of individual modes (Appendix B). In all photon counting measurements, the detector gating width is set to 10 ns and the deadtime is equal to the clock period. The InGaAs SPDs have dark counts of 5.09 × 10−5 and 3.18 × 10−5 per gate, respectively, at a quantum efficiency of 15% and a clock frequency of 250 kHz. The after-pulsing probability is less than 8%.

Fig. 6
Fig. 6

Normalized coincidence spectra for P 2 I . The blue and green curves show the cases with different external coupling conditions and thus different averaged photon lifetimes. The gray curve shows the instrument response function (IRF) of our coincidence counting system, with a FWHM of 312 ps which primarily come from the timing jitters of the two InGaAs SPDs. The red and purple curves show the theoretical prediction. The coincidence spectra were recorded with a detector gating width of 10 ns, a quantum efficiency of 25%, and a clock frequency of 125 kHz. The coincidence counter has a time resolution of 4 ps.

Fig. 7
Fig. 7

Schematic of photon pair generation in a silicon microdisk resonator.

Fig. 8
Fig. 8

Schematic of photon pair generation in a silicon microdisk resonator. The clockwise and counter-clockwise modes are coupled inside the cavity.

Fig. 9
Fig. 9

a. Schematic of coincidence photon counting. A signal and idler photon arrive at the detectors at time ts and ti, respectively, with a probability density of pph(ts, ti). The two detectors produce a pair of events which are recorded by the coincidence counter at time t1 and t2, with a probability density of p(t1, t2). b. The time relationship of the events arriving at the coincidence counter. Each event arrives within a gating window centered at tj0 (j = 1, 2) and with a width of TG. The two input channels of the coincidence counter has a time delay of Td.

Fig. 10
Fig. 10

Schematic of the experimental setup (details can be found in Fig. 2).

Fig. 11
Fig. 11

Photon fluxes recorded for P 2 I , with detector dark counts subtracted. The solid and open circles show the photon fluxes when the center wavelengths of the bandpass filters are tuned onto and away from the P 2 I mode pair, respectively, with the tapered fiber touched down to the nanoforks. The open triangles are recorded when the tapered fiber is lifted up and thus does not couple to the device. The blue and red show the signal at shorter wavelength and idler at longer wavelength, respectively. The dashed lines are only for eye guidance.

Fig. 12
Fig. 12

a. Raman noise spectra recorded with different lengths of fiber 1, with an input pump power of about 270 μW. The spectra were recorded at different transmission ports of the CWDM DEMUX (indicated by color), for easy suppression of the pump. The solid curves and dotted curves show the cases with a fiber length of ∼29 m and ∼1 km, respectively. The amplitudes of the dotted curves were reduced by a factor of 10, for easy comparison with the thin solid curves. The inset shows the case with a fiber length of ∼5 m, same as what we used to characterize correlated photon pairs. The pump power and wavelength are the same for these three cases. The spectral cut-off beyond ∼1590 nm is due to the quantum-efficiency cut-off of the detector used in our sepctrometer. b. Raman noise spectra with the pump located at 1470 nm, showing the Stokes side. fiber 1 has a length of ∼1 km. c. Raman noise spectra with the pump located at 1570nm, showing mostly the anti-Stokes side. Fiber 1 has a length of ∼1 km.

Equations (43)

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p c ( t s , t i ) = Γ es Γ ei Γ ¯ 2 ( g N p ) 2 e Γ tj | t s t i | ,
R c = Γ es Γ ei Γ ts Γ ti 2 ( g N p ) 2 Γ ¯ .
H 0 = j = p , s , i { ω 0 j a j a j Γ ej [ a j b j e i ω j t + b j a j e i ω j t ] } ,
H I = 2 g p ( a p ) 2 a p 2 2 a p a p ( g ps a s a s + g pi a i a i ) g psi a s a i a p 2 g psi * ( a p ) 2 a s a i ,
g ijkl 3 ω 0 i ω 0 j ω 0 k ω 0 l η ijkl 4 ε 0 n i n j n k n l V ¯ ijkl χ ( 3 ) ( ω 0 i ; ω 0 j , ω 0 k , ω 0 l ) ,
V v = { d r ε r ( r , ω 0 v ) | E ˜ v ( r , ω 0 v ) | 2 } 2 si d r ε r 2 ( r , ω 0 v ) | E ˜ v ( r , ω 0 v ) | 4 ,
η ijkl si d r ( ε r i ε r j ε r k ε r l ) 1 / 2 E ˜ i * E ˜ j E ˜ k * E ˜ l { Π v = i , j , k , l si d r ε r v 2 | E ˜ v | 4 } 1 / 4 ,
g p g pj g psi g = c η n 2 ω p ω s ω i n s n i V ¯ ,
d a p d t = ( i ω 0 p Γ tp / 2 ) a p + i g a p a p 2 + i Γ ep b p e i ω p t + i Γ 0 p u p ,
d a s d t = ( i ω 0 s Γ ts / 2 ) a s + 2 i g a p a p a s + i g a i a p 2 + i Γ es b s e i ω s t + i Γ 0 s u s ,
d a i d t = ( i ω 0 i Γ ti / 2 ) a i + 2 i g a p a p a i + i g a s a p 2 + i Γ ei b i e i ω i t + i Γ 0 i u i ,
f j = b j + i Γ ej a j .
f s ( t ) f s ( t ) = Γ es Γ ti 2 π + | B ( ω ) | 2 d ω ,
f i ( t ) f i ( t ) = Γ ei Γ ts 2 π + | B ( ω ) | 2 d ω ,
p c ( t s , t i ) f i ( t i ) f s ( t s ) f s ( t s ) f i ( t i ) f s ( t s ) f s ( t s ) f i ( t i ) f i ( t i ) = Γ es Γ ei | 1 2 π + B ( ω ) [ Γ ts A ( ω ) 1 ] e i ω ( t s t i ) d ω | 2 ,
A ( ω ) = Γ ti / 2 i ω ( Γ ts / 2 i ω ) ( Γ ti / 2 i ω ) ( g N p ) 2 ,
B ( ω ) = i g a p 2 ( Γ ts / 2 i ω ) ( Γ ti / 2 i ω ) ( g N p ) 2 ,
f s ( t ) f s ( t ) 2 Γ es ( g N p ) 2 Γ ts Γ ¯ ,
f i ( t ) f i ( t ) 2 Γ ei ( g N p ) 2 Γ ti Γ ¯ ,
p c ( t s , t i ) { Γ es Γ ei Γ ¯ 2 ( g N p ) 2 e Γ ts ( t s t i ) if t s t i Γ es Γ ei Γ ¯ 2 ( g N p ) 2 e Γ ti ( t i t s ) if t s < t i .
R c = Γ es Γ ei Γ ts Γ ti 2 ( g N p ) 2 Γ ¯ ,
H 0 = j = p , s , i { ω 0 j ( a jf a jf + a jb a jb ) ( β j a jf a jb + β j * a jb a jf ) Γ ej [ ( a jf b jf + a jb b jb ) e i ω j t + ( b jf a jf + b jb a jb ) e i ω j t ] } ,
H I = g p 2 [ ( a pf ) 2 a pf 2 + ( a pb ) 2 a pb 2 + 4 a pf a pf a pb a pb ] 2 ( a pf a pf + a pb a pb ) [ g ps ( a sf a sf + a sb a sb ) + g pi ( a if a if + a ib a ib ) ] g psi [ a sf a if a pf 2 + a sb a ib a pb 2 ] g psi * [ ( a pf ) 2 a sf a if + ( a pb ) 2 a sb a ib ] ,
d a pf d t = ( i ω 0 p Γ tp / 2 ) a pf + i β p a pb + i g ( a pf a pf + 2 a pb a pb ) a pf + i ζ pf ,
d a pb d t = ( i ω 0 p Γ tp / 2 ) a pb + i β p * a pf + i g ( a pb a pb + 2 a pf a pf ) a pb + i ζ pb ,
d a sf d t = ( i ω 0 s Γ ts / 2 ) a sf + i β s a sb + 2 i g ( a pf a pf + 2 a pb a pb ) a sf + i g a if a pf 2 + i ζ sf ,
d a sb d t = ( i ω 0 s Γ ts / 2 ) a sb + i β s * a sf + 2 i g ( a pf a pf + a pb a pb ) a sb + i g a ib a pb 2 + i ζ sb ,
d a if d t = ( i ω 0 i Γ ti / 2 ) a if + i β i a ib + 2 i g ( a pf a pf + a pb a pb ) a if + i g a sf a pf 2 + i ζ if ,
d a ib d t = ( i ω 0 i Γ ti / 2 ) a ib + i β i * a if + 2 i g ( a pf a pf + a pb a pb ) a ib + i g a sb a pb 2 + i ζ ib ,
f s ( t ) f s ( t ) = Γ es Γ ti 2 π + [ | M 13 ( ω ) | 2 + | M 14 ( ω ) | 2 ] d ω ,
f i ( t ) f i ( t ) = Γ ei Γ ts 2 π + [ | M 31 ( ω ) | 2 + | M 32 ( ω ) | 2 ] d ω ,
p c ( t s , t i ) f i ( t i ) f s ( t s ) f s ( t s ) f i ( t i ) f s ( t s ) f s ( t s ) f i ( t i ) f i ( t i ) Γ es Γ ei | 1 2 π + { Γ ts [ M 11 ( ω ) M 31 * ( ω ) + M 12 ( ω ) M 32 * ( ω ) ] M 31 * ( ω ) } e i ω ( t s t i ) d ω | 2 ,
M 1 ( ω ) = ( i ω Γ ts / 2 i β s i g a pf 2 0 i β s * i ω Γ ts / 2 0 i g a pb 2 i g * ( a pf * ) 2 0 i ω Γ ti / 2 i β i * 0 i g * ( a ab * ) 2 i β i i ω Γ ti / 2 ) .
p SI ( t 1 , t 2 ) = ξ 1 ξ 2 + p ph ( t s , t i ) p J ( τ 1 , τ 2 ) d τ 1 d τ 2 = ξ 1 ξ 2 + p ph ( t 2 t 1 τ D ) p IRF ( τ D ) d τ D ,
p IRF ( τ D ) 0 + p J ( τ 1 , τ 1 + τ D ) d τ 1 .
p ( t 1 , t 2 ) = p SI ( t 1 , t 2 ) + p SD ( t 1 , t 2 ) + p DI ( t 1 , t 2 ) + p DD ( t 1 , t 2 ) ,
p ( t 1 , t 2 ) = p SI ( t 2 t 1 ) + p s ( t 1 ) p D 2 ( t 2 ) + p D 1 ( t 1 ) p I ( t 2 ) + p D 1 ( t 1 ) p D 2 ( t 2 ) .
ρ ( τ ) d τ = t 10 T G / 2 t 10 + T G / 2 d t 1 t 20 T G / 2 t 20 + T G / 2 d t 2 p ( t 1 , t 2 ) = d τ ( T G | τ T d | ) [ p SI ( τ T d ) + p S p D 2 + p D 1 p I + p D 1 p D 2 ] ,
C ( τ ) d τ = R G T ρ ( τ ) d τ ,
ρ ( τ ) d τ = d τ T G [ p SI ( τ T d ) + p S p D 2 + p D 1 p I + p D 1 p D 2 ] ,
C T ( τ ) d τ = R G T d τ ( T G | τ T d | ) [ p SI ( τ T d ) p S p I ] = R G T d τ ( T G | τ T d | ) ξ 1 ξ 2 α s α i + p c ( τ T d τ D ) p IRF ( τ D ) d τ D ,
C A ( τ ) d τ = R G T d τ ( T G | τ T d | ) ( p S + p D 1 ) ( p I + p D 2 ) ,
CAR = T FWHM / 2 + T FWHM / 2 C T ( τ + T d ) d τ T FWHM / 2 + T FWHM / 2 C A ( τ + T d ) d τ ,

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